Singular and Degenerate Cauchy Problems ACADEMIC PRESS RAPID MANUSCRIPT REPRODUCTION

This is Volume 127 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, University of Southern California

The complete listing of books in this series is available from the Publisher upon request. Singular and Degenerate Cauchy Problems

R. w. Ccxrroll Department of Mathematics University of Illinois at Urbana-Champaign

R. E. Showalter Department of Mathematics University of Texas at Austin

ACADEmIC PRESS New York San Francisco London 1976

A Subsidiary of Harcourt Brace Jovanovich, Publishers COPYRIGHT0 1976. BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PIIRMISSION IN WRITING FROM THE PUBLISHER.

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Library of Congress Cataloging in Publication Data

Carroll, Robert Wayne, Date Singular and degenerate Cauchy problems. (Mathematics in science and engineering ; ) Bibliography: p. Includes index. 1. Cauchy problem. I. Showalter, Ralph E., joint author. 11. Title. Ill. Series. QA 3 74.C 38 5 15’.353 76-46563 ISBN 0-12-161450-6

PRINTED IN THE UNITED STATES OF AMERICA Contents

PREFACE vi i 1 Singular Partial Differential Equations of EPD Type 1 .l. Examples 1 1.2. Mean values in R" 5 1.3. The Fourier method a 1.4. Connection formulas and properties of solutions for EPD equations in D'and S' 29 1.5. Spectral techniques and energy methods in Hilbert space 43 1.6. EPD equations in general spaces 66 1.7. Transmutation 80 2 Canonical Sequences of Singular Cauchy Problems 2.1. The rank one situation a9 2.2. Resolvants 96 2.3. Examples with mzo,= 0 101 2.4. Expressions for general resolvants 130 2.5. The Euclidean case plus generalizations 136

V CONTENTS

3 Degenerate Equations with Operator Coefficients 3.1. Introduction 143 3.2. The Cauchy problem by spectral techniques 147 3.3. Strongly regular equations 166 3.4. Weakly regular equations 179 3.5. Linear degenerate equations 187 3.6. Semilinear degenerate equations 206 3.7. Doubly nonlinear equations 229

Selected Topics 4.1. Introduction 237 4.2. Huygens' principle 237 4.3. The generalized radiation problem of Weinstein 245 4.4. Improperly posed problems 251 4.5. Miscellaneous problems 264

REFERENCES 215

INDEX 329

vi Prefoce

In this monograph we shall deal primarily with the Cauchy problem for singular or degenerate equations of the form (u' = u, = au/at) (1.1) A(t)u,, + BWu, + C(t)u=g where 4-1 is a function of t, taking values in a separated locally convex space E, while A(t),B(t), and C(t)are families of linear or nonlinear differential type operators acting in €, some of which become zero or infinite at t = 0. Appropriate initial data u(0) and u,(O) will be prescribed at t=O, andg is a suitable €-valued function. Similarly some equations of the form (1.2) (A(-)u),,+ (B(.)u),+ C(.)U =g will be considered. Problems of the type (1.1) for example will be called singular if at least one of the operator coefficients tends to infinity in some sense as t + 0. Such problems will be called degenerate if some operator coefficient tends to zero as t+ 0 in such a way as to change the type of the problem. Similar considerations apply to (1.2). We shall treat only a subclass of the singular and degenerate problems indicated and will concentrate on well-posed problems. If (1.1) or (1.2) can be exhibited in a form where the highest order derivative appears with coefficient one, then the problems treated will be of parabolic or hyperbolic type; if the highest order derivative cannot be so isolated, the equation will be said to be of Sobolev type. We have included a discussion of not necessarily singular or degenerate Sobolev equations for completeness, since it is not available elsewhere. The distinction between singular and degenerate can occasionally be somewhat artificial when some of the operators (or factors thereof) are invertible or when a suitable change of variable can be introduced (see, e.g., Example 1.3); however, we shall not usually resort to such artifices. In practice E frequently will be a space of functions or distributions in aregionclCclR"andO=ztcb<~. The study of singular and degenerate Cauchy problems is partially motivated by problems in physics, geometry, applied mathematics, etc., many examples of which

vii PREFACE are given in the text, and there is an extensive literature. We have tried to give a fairly complete bibliography and apologize for any omissions. The material is aimed basically at analysts, and standard theorems from functional analysis will be assumed (cf. Bourbaki [l;21, Carroll [141, Horvath [ll, Kothe [ll, Schaeffer [ll, Treves [l] 1; in particular Schwartz distributions will be used with the standard notations (cf. Schwartz [l I 1. There is also some material involving the abstract theory of Lie groups which is assumed but clearly indicated, with references, in Chapter 2; and there are enough concrete examples worked out in the text to illustrate the matter completely for the reader unfamiliar with Lie theory. The chapters are essentially independent; Chapters 1 and 3 begin with introductory material outlining content, motivation, and objectives, whereas Chapters 2 and 4 are organized somewhat differently. Some remarks on notation should perhaps be made here. If, in a given chapter, there appears a reference to formula (x.y) it means (x.y) of that chapter. If a reference (x.y. z) appears, it means formula (y. z) of Chapter x. Theorems, Lemmas, Examples, Remarks, etc. will be labeled consecutively; thus one might have in order: Lemma x.y, Theorem x.y + 1, Remark x.y +2, Lemma x.y + 3, etc. in a given section. The first author (R.W.C.) would like to acknowledge a professional debt to A. Weinstein, who initiated systematic work on Euler-Poisson-Darboux (EPD) equations (cf. Weinstein [l;21 1 and whose further contributions to this theory have been essential to its development; his encouragement motivated this author's early work in the area. He was also instrumental in the undertaking of the present monograph. Thus this book is dedicated to Alexander Weinstein. The authors would like to acknowledge the aid of the National Science Founda- tion from time to time in supporting some of the research presented here.

viii Chapter 1

Singular Partial Differential Equations of EPD Type

1.1 Examples. Let us first give some typical examples of singular and related degenerate problems; their solutions , in various spaces via diverse techniques, will appear in the course of the book, usually as special cases of more general results.

Further examples of degenerate problems will appear in Chapter 3.

Example 1.1 The singular Cauchy problem for EPD equations.

Let t 2 0 and x E Rn; we consider the problem

m 2m+lUm m (1.1) u . + tt -tt

(1.2) Urn(X,O) = f(x); U!(X,O) = 0 where the parameter m E C is arbitrary for the moment, Ax denotes the Laplace aperator in Rn , and classically f and um were numerical functions (note that the left hand side of (1.1) denotes the radial form of a Laplace operator in Rn if 2m + 1 = n - 1). These equations, for integral m, provide a model for some very interesting scales of canonical singular Cauchy problems with their natural origins in Lie theory (see Chapter 2).

They also arise in physics as is indicated in Example 1.3 and of course for m = -1/2 we have the wave equation; their connection with mean values is indicated in Section 2 and in Chapter 2. In various forms one can trace their origins back to Euler [l],

Poisson 111, and Darboux [l]. In this chapter we will present

1 SINGULAR AND DEGENERATE CAUCHY PROBLEMS five general techniques for the solution of these problems and of similar abstract singular problems; more general singularities in the ut term and some nonlinear terms will be treated. The first is based on a Fourier method in distribution spaces developed by one of the authors (cf. Carroll [2; 3; 4; 5; 6; 8; 10; 11; 141).

The second involves a spectral technique in a Hilbert space related to the Fourier method (cf. Carroll, loc. cit.) while the third is based on transmutation methods in various spaces (cf.

Carroll [24], Carroll -Donaldson [20] , Delsarte [l], Delsarte-

Lions [2; 31 Donaldson [l; 3; 51, Hersh [2], Lions [l;2; 3; 4;

51, Thyssen [l;21). The fourth involves the idea of related differential equations (cf. Bragg [2; 3; 4; 5; 91, Bragg-Dettman [l;

6; 7; 81, Carroll [24], Carroll-Donaldson [20], Dettman [l],

Donaldson [l; 3; 51, Donaldson-Hersh [6], Hersh [l;2; 31). There are certain relations between the method of transmutation and that of related equations and this is brought out in Carroll [24],

Carroll-Donaldson [20], and Hersh [2] (cf. also Carroll [18; 19;

251, Donaldson [4; 71, and Hersh [l;21). The fifth technique involves "energy" methods following Lions [5] and yields some results of Lions indicated in Carroll [8].

Remark 1.2 There are many additional papers devoted to problems of the form (1.1) - (1.2) and we will list some of them here; further references can be found in the bibliographies to these articles. Some of this material will appear in this or other chapters. In giving such references we have tried to avoid

2 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE repetition of the citations above and there will be considerable relevant material and further references in the work on the Cauchy problem for abstract Tricomi type problems to be treated later in Chapter 3 (cf. Example 1.3), in Remark 2.4 on mean values, and in Chapter 4. Thus, let us cite here Agmon [2], Babenko [l],

Baouendi-Goulaouic [l], Baranovskij [l; 2; 3; 4; 5; 6; 71, Barantsev [l], Berezanskij [l], Berezin [l], Bers [l; 21, Bitsadze [l; 21, Blumkina [l], Bresters [l], Bureau [l; 21, Carroll [7; 9; 13; 18; 19; 21; 221, Carroll-Silver [15; 16; 171, Carroll-Wang [I2], Chi [I; 21, Cibrario [l; 2; 31, Cinquini-Cibrario [1;2], Conti [1 ; 2; 31, Copson [l], Copson-Erdelyi [Z], Courant-Hi1 bert [l], Delache-Leray [l], Diaz [5; 61, Diaz-Kiwan [7], Diaz-Ludford [3; 41, Diaz-Martin [9], Diaz-Weinberger [2], Diaz-Young [l; 81, Ferrari-Tricomi [l], Filipov [l], Fox [l], Frank'l [l], Fried- lander-Heins [l], B. Friedman [l], Fusaro [2; 3; 4; 5; 61, Gara- bedian [l], Germain [l], Germain-Bader [2; 31, Gilbert [l], Gordeev [l], Gcnther [2; 31, Haack-Hellwig [1], Hariullina [l; 2; 31, Hairullina-Nikolenko [l], Hellwig [l; 2; 31, Kapilevi?! [l; 2; 3; 4; 5; 61, Karapetyan [l], Karimov-Baikuziev [l; 23, Karmanov [l], Karol [l; 21, Kononenko [l], Krasnov [l; 2; 31, Krivenko [l; 23, Ladygenskaya [l], Lavrentiev-Bitsadze [l], Lieberstein [l; 21, Makarov [l; 21, Makusina [l], Martin [l], Mikhlin [l], Miles-Williams [l], Miles-Young [Z; 31, Morawetz [l; 21, Nersesyan [l], Nevostruev [l], Nosov [l], Olevskij [2], Ossicini [l; 21, Ovsyanikov [l], Payne [1], Payne-Sather [Z], Protter [2; 3; 4; 5;

3 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

61, Protter-Weinberger [1], Rosenbloom [l; 2; 31, Sather-Sather

[4], Silver [l],Smirnov [l;2; 3; 41, Solomon [l;21, Sun [l], Suschowk [l],Tersenov [l;2; 3; 41, Tong [l],Travis [1], Tri- comi [1], Volkodarov [l],Walker [l;2; 31, Walter [l],Wang [l;

21, Weinstein [l;2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 13; 14; 15; 16;

17; 18; 19; 203, Williams [l],Young [l;2; 3; 4; 5; 6; 7; 8; V 9; 10; 111, Zitomirskij [l].

Example 1.3 The degenerate Cauchy problem for the Tricomi equation. The original Tricomi equation is of the form

(1.3) Utt - tuxx = 0 and we will be primarily concerned with the Cauchy problem for abstract versions of the more general equation

(1.4) u tt + a(x,t)ut + b(x,t)u, + c(x,t)u - K(x,t)uxx = 0 in the hyperbolic region t > 0 where it is assumed that tK(x,t) >

0 for t p 0 while K(x,O) = 0; suitable initial data u(x,O) and u (x,O) will be prescribed on the parabolic line t = 0. This ex- t ample will be treated as a degenerate problem in Chapter 3 but is displayed here because of the connection of (1.3) to a (singular)

EPD equation of index 2m + 1 = 1/3 under the change of variable

T = 2/3 t3l2for t > 0 which yields

I (1.5) UTT + 3-r UT = uxx

Similarly in the region t < 0 the change of variable ~=2/3(-t)3/2

4 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1 transforms (1.3) into uTT + -3* u ~ + uxx = 0 which is a singular elliptic equation of the type studied by Weinstein and others in the context of generalized axially symnetric potential theory

(GASPT); cf. Remark 1.2 for some references to this, The references above in Remark 1.2 also deal with mixed problems where suitable data is prescribed on various types of curves in the elliptic (t < 0) and hyperbolic (t > 0) regions; such problems occur for example in transonic gas dynamics. We will not deal with the elliptic region in general but some other types of singular hyperbolic problems for EPD equations will be discussed briefly in

Chapter 4.

1.2 Mean values in Rn . We refer now to the EPD problem (1.1) - (1.2) for x E Rn and 2m + 1 = n - 1 which is a pivotal case in the theory. Let

n where 0 E E, = 1 xi, wn = 2rn"/r(n/2) is the surface area 1 of the n-dimensional un t ball, and du = tn- dRn represents the n surface area measure on the ball of radius Ix = t. The surface mean value measure 1-1 (t) averages 41 over a sphere of radius t cen- X tered at the origin. Similarly one defines a solid mean value

5 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

I operator Ax(t) E E (t still fixed) by the rule

which averages @ E E over the entire ball 1x1 5 t. Now we observe that =

and, allowing t to vary, the Y following calculations are easily checked (v denotes the exterior norma 1 )

r -1 (%)don = 1 A@dx n

t t = = -n -n

Note in (2.3) that if E D with I) = 1 in a neighborhood of

I 1x1 5 t then the E-E pairing , in a I D-D pairing so that D = , =

By Lemma 2.3 below, (2.3) and (2.4) imply that the functions t -+ px(t) and t -f Ax(t) belong to C"(Ei) for t > 0 with

(2.5) xpx(t)d = ;t Aflx(t)

6 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Theorem 2.1 Let T E Di be arbitrary. Then u:(t) = px(t) * T satisfies the EPD equation (1.1) with 2m + 1 = n - 1 n (i.e., m = 1) and the initial conditions um(o) = T with - X P(o) = 0. s!-dt x

I I I

Proof: We recall that (S, T) + S * T: E x D -+ D is separately continuous so equation (1 .l)results immediately by differentiating (2.5) and using (2.5) - (2.6). Evidently

px(t) -+ 6 (Dirac measure at x = 0) when t -+ 0 and we note that dtd (px(t) * T) = ;t AAx(t) * T = (:)[A6 * Ax(t) * TI = t (F) (Ax(t) * AT) -+ 0 as t -+ 0. QED

For information about distributions and the d f ferent ia tion of vector valued functions let us refer to Carroll [141, A. v Friedman [Z], Garsoux [l],Gelfand-Silov [5; 6; 71 Horvlth [l], Nachbin [l],Schwartz [l; 21, Treves [l].

Definition 2.2 A function f:R -+ E, E a general locally I I* convex space, has a scalar derivative fs(t)E (E ) if d I I I I I* -f(*),dt e > = for all e E E (F denotes the alge- df I braic dual of F). On the other hand if = f (t) E E in the I topology of E then f is called the strong derivative of f.

The following lemma is a special case of results in Carroll

[2; 5; 141 and Schwartz [2].

7 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

I I Lema 2.3 If f:R + E has a scalar derivative fSthen f is I I strongly differentiable with f = fs.

f ( t ) - f ( to> Proof: Fix to and set g(t) = - to . In discussing limits as t -t to it suffices to consider sequences tn +. to since any to has a countable fundamental system of neighborhoods. By I hypothesis g(tn) +. fs(to)weakly. Using the fact that E is a

Monte1 space and a version of the Banach-Steinhaus theorem (cf.

Carroll [14]) it follows that g(tn) +. f;(to) strongly in E' so I I that fs(to)= f (to). QED

Remark 2.4 A great deal of information is available relating

"spherical" mean values and differential equations, both in Rn and on certain Riemannian manifolds. In addition to references included in the Introduction, Example 1.1, Remark 1.2, and work on GASPT, relative to such mean values, let us mention Asgeirsson

[l],Carroll [l],Fusaro [l],Ghermanesco [l;21, Ginther [l],

Helgason [l;2; 3; 4; 5; 6; 7; 8; 9; 10; 111, John [l],Olevskij

[l],Poritsky [l],Walter [2], Weinstein [12], Willmore [l]and their bibliographies. This will be discussed further in a Lie group context in Chapter 2. For some references to complex function theory and various mean values and measures related to differential equations see Zalcman [l;21 and the bibliographies there.

1.3 The Fourier method. We will work with a more general equation than (1.1) which includes (1.1) as a special case. Thus

8 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

where the aa are constant (and real) and the sum in (3.1) is over a finite set of mu1 ti-indices a = (a1 . . ., an). The Fourier A I I transform F:T -+ T = FT:S -+ S is a topological isomorphism (cf.

Schwartz [l]) and for $ E S for example we use the form F$ = A fm n $(XIexp - idx where = 1 x y Hence in par- $ = -m i i' 1 A ticular F(Dk$) = ykF$ = y k $. We recall also that AxT = A 6 * T 8l with F6 = 1 and will write A(y) = F(Ax6.) = 1 aaya = 1 aayl . . . an yn . It will be assumed that A(y) satisfies

Condition 3.1 A(y) = 1 aaya is real with A(y) 2 a > 0 for

IYI > Row In particular Condition 3.1 holds when Ax = -A = -1 i 22-D - X k 2 1 0; so that F(-A6) = 1 yk = lyI2. It can also hold for certain hypoelliptic operators A since then IA(y)l -+ as IyI -+ (cf. Hb'rmander [l]) or even nonhypoelliptic operators (e.g., A(y) = 2 2 a + y2 + . . . + yn). One might also envision a parallel theory when Re A(y) 2 a > 0 for IyI > Ro but we will not investigate 2' this here. Let us first consider the problem for ~"(0) E C (Sx) on 0 5 t 5 b < (b arbitrary)

9 SINGULAR AND DEGENERATE CAUCHY PROBLEMS where m 2 -1/2 is real (complex m with Re m > -1/2 can also be treated by the methods of this section). Taking Fourier transforms in x one obtains

(3.4) + A(y) = 0 imtt + Eiyt im "m "m (3.5) w (0) = T; W&O) = 0 and when T = 6 we denote the solution of (3.2) - (3.3) by Rm(t) h so that $(t) satisfies (3.4) - (3.5) with T = 1. The expressions

Rm(t) and im(t)will be called resolvants and one notes that if

Rm(t) E 0; (i.e., @(t)E OM), with suitable differentiability in

O;, then Rm(t) * T = wm(t) satisfies (3.2) - (3.3) (see Theorem

3.4). We recall here that OM is the space of COD functions f on

Rn such that Daf is bounded by a polynomial (depending perhaps on a) while fs -+ 0 in OM whenever gD b + 0 uniformly in IRn for a6 any a and any g E S. 0; = F-'OM C SI and is given the topology I transported by the Fourier transform (i.e., T6 -+ 0 in Oc c-f

FT6 -+ 0 in OM). It is known that Oc is the space of distributions

I I mapping S -+ S under convolution and the map (S,T) + S * T: I I I

Oc x S -+ S is hypocontinuous.

A Now the solution of (3.4) - (3.5) when T = 1 is given by

("R"(t) = F(.,t))

10 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE where z = t (cf. B. Friedman [l] and Rosenbloom [l]). One observes that $(y,t) is in fact a function of z2 so that it is not necessary to define the square root m; evidently R^m(-,t) n is an entire function of the yk for yk E E (and hence of y E (I: by Hartog's theorem - cf. Hzrmander [Z]).

"m Lemma 3.2 The funct on t + R (-,t):[o,b] -+ OM is cont nuous (m 2 -1/2).

Proof: Clearly t + $(-,t):[o,b] +. E is continuous and one Y knows that on bounded sets in OM the topology of O,., coincides with that induced by E (see Schwartz [2] and recall that a set B C OM is bounded if Daf is bounded by the same polynomial , depending perhaps on a, for all f E B). Consequently one need only show "m that each DaR (y,t) is bounded by a polynomial in y, independent of t for t E [o,b]. We choose here a straightforward method of doing this and refer to subsequent material for another, much more general , technique (cf. Carroll [5; 81). Thus, one knows that for m > -1/2 (cf. Watson [l])

m = z '"5 = z). (the case -1/2 is trivial since -1 /2 (z) (2/.rr)'" cos

Writing la1 = a, + . . . + an and c, = 2r(m+l)/JiT r(m + -!-)2 we have from (3.7)

11 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Now for IyI > R we write 0 a Am 1 a “m (3.9) -R (y,t) = (tL/2m) 57 R (y,t) ayk ayk and since A(y) 2 a with aA(y)/ayk = Bk(y) a polynomial we have from (3.8) - (3.9)

I aim(y,t) I xbcm 22 (3.10) -lBk(Y)I = clBk(Y)I 5 1 + c Bk(Y) ayk 46 A similar argument obviously applies to majorize any D RIn (y,t) for a IyI > Ro by a polynomial in y not depending on t. For IyI 5 Ro

the power series in (3.6) may be used, since (y,t) -+ Daim(y,t) is continuous on the compact set (lyl 5 Ro) x [o,b], to obtain a In bound IDaim(y,t,)l 5 Ma. Consequently lDaR (y,t) I 5 HIa + polynomial = polynomial in y, for all y, independent of t E [o,b].

QE D

Lenma 3.3 The function t -+ ^Rm(-,t) belongs to Cm(O,,) for t E [o,b] (m 2 -1/2) and

12 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Proof: The formulas (3.11) - (3.12) are easily seen to hold in E by using well-known recursion formulas for Bessel functions Y (cf. Watson [l], Carroll [5]). Further we know by Schwartz [3;

41 that if t +. ?(*,t)E Co(OM) f'l C1(E) with t +. iT(*,t)E Co(OEl) % 1 then t +. R (-,t)E C (Old). Using (3.11) and Lemma 3.2 however we "m see that t +. Rt(*,t) E Co(OM) and in particular (3.11) - (3.12) hold in OM. By iteration of this argument it is seen that t +.

"Rm(*,t)belongs to Cm(On). QED

-1 "m Theorem 3.4 Let T E S' and R"(t) = F R (*,t)with i'"(y,t) given by (3.6). Then Rm(t) E 0; and wm(t) = Rm(t) * T satisfies (3.2) - (3.3).

Proof: We know t +. i'(*,t)?:[o,b] + SI is continuous since

AA Ah I I the map (S,T) +. ST:OM x S +. S is hypocontinuous (cf. Schwartz 2m t lAm [l]). Using (3.11), t + Rt(-,t)?:[o,b] +. SI is also continuous. From (3.4) in E we see (using (3.11) again) that t +.

A A RFt(-,t) = A(*)[W'(-,t) - Rm(=,t)] belongs to Co(OM) 2 which shows explicitly that t + im(-,t)E C (OM) on [o,b] (cf.

Lemma 3.3) and that t +. $(*,t)?satisfies (3.4) in SI. Conse- quently (3.2) holds in SI and wT(o) = 0 since by (3.11) $(y,o) =

0. QE D

To deal with uniqueness we make first a few formal calculations following Carroll [8; 111; this technique is somewhat different than that used in Carroll [5] but by contrast it can be A generalized. Thus for o < T 5 t 5 b < m let Rm(y,t,T) and

13 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

^sm(y,t,-r) be two linearly independent solutions of (3.4) with "m "m "m R (Y,T,T) = 1, Rt(y,~,.r) = 0, S (~,T,T) = 0, and $(y,-r,-~) = 1.

Such (unique) numerical solutions exist since the problem is nonsingular on [~,b] for t > 0. Let

SO that ^cm is a fundamental matrix for the first order system corresponding to (3.4) written in the form

A (3.14) + ip = 0; M(y,t) =

As in Schwartz [3] it is easily shown that

from which follows in particular

Given that ^Rm and irnhave suitable properties (stated and verified below) it follows that if is a solution of (3.4) in S' "m A with w (T) = T and ;"(T) = 0 then using (3.16) (3.17) the for- t - mulas (3.18) - (3.20) below make sense.

14 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

t im(t)- ^mR (y,t,-c)? = im(t)- "mR (y,t,T)?

From (3.20) it then follows formally that any solution 2 of I "m A ^m (3.4) in S with w (T) = T and w~(T)= 0 (T > 0) must be of the form im(t) = im(yytyT)?. We will now proceed to justify this and to attend to the case T = 0 as well. We remark also that if one considers

A with initial conditions given at t = T as above(f(-)cCo(SI)) then the procedure of (3.20) formally yields the unique solution as

(3.22) $(t) = im(yytyT)?+ r?(y,t,<)?(f)d< T

We will develop a procedure now which can also be used in other situations (see Section 1.5). It is helpful to have an explicit example of the technique however and hence we introduce it here; in particular this will make the considerable details more easily visible and shorten the development later. Thus for m 2 - 1/2 let P(t) = exp (-ry d5) = (t/b)2m+1 so that P(T)/P(t) = t (-c/t)2m+1 5 1 for 0 S T 5 t 5 b and P(t) -+ 0 as t -+ 0. Let us consider equation (3.4) for km(y,t,T) with initial data at

15 SINGULAR AND DEGENERATE CAUCHY PROBLEMS t = T > 0 as in (3.13) and then, multiplying by P(t), we obtain

(3.23) & (Pi!) + PA^Rm = 0

This leads to the integral equation (T 5 5 5 u 5 t), equivalent to (3.4), with initial conditions at t = T as indicated,

and evidently T = 0 is permitted in (3.24). A solution to (3.24) is given formally by the series

“m 03 (3.25) R (y,t,-c) = 1 (-l)kJk 1 k=O

(cf. Hille [l]) where Jk denotes the kth iterate of the operation

J and Jo is the identity. Writing imYo= 1 and

(3.26) imYp= 1 - = f (-l)kJk 1 k=O we have R - ^RmYP-’ = (-1)’Jp-1 and if y E KC Cny with K compact, then IA(y)IC 5 cK and in particular

5 cKF(t,-c) 5 cKF(t,o) = cKF(t) where F(t, T) is the double integral in (3.27) and F(tl) 5 F(t) 5 t2 F(b) for 0 S t, 5 t i b (note that F(t) = Continuing we have, using (3.27) ,

16 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPDTYPE

h .., 22 so that - Rmy' Ic 5 cKF (t)/2! since if F(t) = -1 - 1: F(a) ($)2m+1dSdo then F = F F' with F(o) = F(o) = 0. By 0 iteration one obtains

Hence the series (3.25) converges absolutely and uniformly on

[O 5 T 5 t S bl x K and since the terms Jp 1 are continuous

h in (y,t,T) and analytic in y the same is true of Rm(y,t,-r). It ^m is clear that R given by (3.25) satisfies (3.24) and we can state

Lemma 3.5 The series (3.25) represents a solution of (3.24) "m "m with R (Y,T,T) = 1 and Rt(y.~,-r) = 0 (0 5 T 5 t 5 b and m > -1/2). "m "m The maps (y,t,-c) + R (y,t,T) and (y,t,-c) -t Rt(y,t,T) are contin- "m "m uous numerical functions while R (*,t,~) and Rt(*,t,.r) are analytic.

Proof: Everything has been proved for imand the statements for ty follow immediately upon differentiating (3.24) in t. QED

We examine now (cf. (3.24))

17 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

For T > 0 there is obviously no problem in proving continuity and

analytic ty as in Lemma 3.5 so we consider T -+ 0. Let

(3.31)

Clearly t,T) -f H(t,.r) is not continuous as (t,T) -+ (0,O) so 2m+l "m there is no hope that (y,t,i-) * et(y.t,.r) will be continuous on [O s T 6 t 5 b] x K (cf. Lemma 3.5). We write H(t,o) = H(t) = zo and then clearly, from (3.30) - (3.31), (y,t) -+

i:(y,t,o) is continuous on [o,b] x K with limit as (y,t) -+

(yoyo) equal to -A(yo).m2m + 1 Further from (3.30) y -+ 2m+l "m R (y,t,-c) is analytic for 0 T i t 5 b. These properties 7 t <- "in are transported to Rtt(y,t,T) after differentiating (3.24) twice

in t and equation (3.4) is satisfied for 0 <- T <- t 5 b. Summar-

izing we have

"m Lemna 3.6 For [O 5 T 5 t 5 b] the maps (y,t) -f Rtt(yytyT) 2m+l "m "m and (y,t) * -7 Rt(y,t,T) are continuous while y + Rtt(y,t,T) and y * $(y,t,r) are analytic. The numerical function -"+'t im(yytyT)satisfies (3.4) with im(y,~,~)= 1 and "mR&Y,T,T) = 0.

Next we will derive some bounds for im,assuming that y is real. If we multiply (3.23) (with t replaced by c) by *m P-'(S)Rg(y.S.r) and integrate by parts it follows that

(3.32) li~(yytyT)12+ 2(2m+l) 15-t 1 ~i~(y,~,T)l2 d~ T + A(y) lkm(y,t,T) = A(Y)

18 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

d "m"m d "m"m since - = 2R Rt and - = 2RtRtt (cf. also Section dt dt lq12 "m 1.4). In particular if IyI > Ro we can say that IRt(yytyT)/25 "m "m A(y) and I^Rm(y,tyT)12 5 1. Since IR (yyty.r)l and IRt(YytyT)l can be bounded on [O 5 T 5 t 5 b] x to 5 IyI 5 R } by continuity 0 (cf. Lemma 3.5) it follows that [im(yytyT)12 5 c1 and "m IRt(yytyT)12 5 c2 + A(y) for 0 5 T 5 t 5 b and all y. To bound the term ?$yytyT) we refer to (3.30) (3.31) to obtain -"+'t -

For IyI > Roy IA(y)l = A(y) and for IyI ;Roy IA(y)I is bounded; this leads to

"m ^m Lemma 3.7 The function R satisfies IR (yytyr)125 cl, "m 2m+l "m IRt(uytyT)12 5 c2 + A(y), and FIRt(yyty-c)I5 c4 + c3 A(y) for

0 2 T < tiband all y E Rn.

"m Let us examine now some similar properties of S . If we *m differentiate (3.24) in T there results (since R (y,~,.) = 1)

= -J + A(y)U(tp)

T T 2m where for m 2 -l/2, U(tyT)= -[I2m - (t) 3 is continuous in (tyT) for 0 5 T 5 t <- b (note that U(tyT) -t -T log (T/t) as m + 0).

From (3.16) we have then

19 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(except perhaps where A(y) = 0). A solution of (3.35) is given

03 (3.36) ? = 2 (-l)kJk U k=O and the terms Jk U in (3.36) are continuous in (y,t,-c) and analytic in y with IU(t,-c)l 2 c. It follows by our previous analysis (cf. (3.26) - (3.29)) that the series in (3.36) converges absolutely and uniformly on [0 <- 'c 5 t 5 b] x K (K C Cn being compact). Defining now .? by this series (whether or not

A(y) = 0) there results

m Lemma 3.8 For m 2 -1/2 the expression S (y,t,T) defined by (3.36) satisfies (3.35) and 3.4); it is continuous in (y,t,-c) "m and analytic in y for [0 5 T 5 t 6 b] along with St (y,t,-c) for "m "m 0 < T 5 t 5 b. One has R = A(y)S (i.e., (3.16)) and, for T > T "m "m 0, (3.17) holds while S (Y,T,T) = 0 with St(y,-c,-c) = 1; finally

P(y,t,o) = o for m > -1/2.

Proof: One notes first that the series obtained by term- "m wise differentiation of (3.25) in T is the same as A(y)S (y,t,T) where :m is given by (3.36). Since Jk U = 0 at t = T for k 2

1 with U(T,T) = 0 it follows that &?(y,-c,~) = 0. On the other "m hand S (y,t,o) = 0 for m > -1/2 since U(t,o) = 0 while (3.17) holds for T > 0 sinde imclearly satisfies (3.35) and hence m "m d (since S (y,-c,-c) = 0) ST = UT - - J = U - J But d-c ? T irnT* 2m+l UT(t,-c) = ,-U(t,-c) - 1 and hence for T > 0

20 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONSOF EPD TYPE

(from (3.35) and the definition of J), from which follows "m St(y,~,~) = 1 for T > 0 and the continuity and analyticity properties of $. To check (3.4) one can use (3.17) to obtain for

T>O

which leads to (3.4) upon differentiation in t. QED

"m Now ST is determined by (3.17) and as T -+ 0 a priori "m lim ST may not exist. However let us look first at

m (3.40) +s^m(y,t,r) = 1 (-l)kJk *. U(.,T) k=I) 1 1 T 2m 1 where 7 U(~,T) = $1 - (c) 3 2 for m > 0 and 0 5 T 5 5 5 t 5 b. Hence, as in (3.26) - (3.29), the series in (3.40) converges uniformly and absolutely for 0 5 T 5 t 5 b and y E K C 1 "m Cn. There is little hope that (y,t,r) + -r S (y,t,T) will be I continuous on [0 S T 5 t 5 b] since e.g., - U(t,-r) is not so T 1 continuous (nor is J -r U(-,T), etc.). We can however let T -f 0 in (3.40) for m > 0 to obtain for t > 0

21 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Consequently from (3.17) for

Lemma 3.9 For m > 0 and m = -1/2 the function (y,t) + "m 1 "m ST(y,t,o) = -R (y,t,o) is continuous for 0 5 t b and y E K 2m 5 "m "m while y + ST(y,t,o) is analytic. Further, S (y,t,?) is absolute- % "m ly continuous for m 2 -1/2 with S (y,t,B) - S (y,t,a) = IBq(y,t,-c)dr and for 0 5 T 5 t S b and m 2 -1/2 the expres- Jci, - "m sion Ts,(y,t,T) is continuous in (y9t,-c) and analytic in y.

Proof: The continuity and analyticity indicated for "m "m ST(y,t,o) and TS,(Y,t,T) follow immediately from (3.42) and ^m (3.17). For 0 < ci, 5 B 5 t 5 b the formula S (y,t,B) - S"m (y,t,a) = jb:(y,t,r)dr is obvious and by continuity one can take limits as a + 0. In the cases -1/2 < m 5 0 we note that 1 will have an integrable singularity at T = 0 since - U(t,-c) - T for -1/2 < m < 0 and - log-c for m = 0. QED

A In order to obtain some bounds for Sm when YE Rn we multiply (3.17) (with T replaced by.5) by im(y,t,E) and since d "m 2 "m"m -1s I = 2s S one obtains upon integration d5 5

22 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

By Gronwall's lemma (cf. Section 5 and Carroll [14]) there re-

A sults ISm(y,t,-c)12 5 c5 for 0 5 T 5 t <, b and y E Rn. Conse-

A quently by (3.17) and Lemma 3.7 (rS:(y,t,~)( 5 c6 on the same domain. Finally from (3.38) we have liyl 5 1 + bck"IA(y)l 5 c7 + c8A(y)*

Lemna 3.10 For y E Rn and 0 S T i t i b one has A ISm(y,t,r)12 5 c5 and lTiy(y,t,T)l 5 c7 + c8A(Y)*

A 2 Lemma 3.11 The function t + Rm -,t,T) E C (E ) for 0 5 T Y "m "m t S b while (t,T) + R (-,t,T) and (t,-c)+ Rt(-,t,.r) : Am [0 5 T 5 t 5 b] -+ E,, are continuous. Similarly 5 -+ S (*,t,<)E J A ^m 2 + SSF(=,t,S) E Co(Ey), t -+ S (*,t,S) E C (Ey), (t,S) + "m 1 E Co(E ), and 5 + R (-,t,S) E C (E ) for 0 S 5 r; t i b Y Y A 1 sm(*,t,S) E C (Ey) and (tL) + q(-,t,S) E Co(E Y ) for 0 < C05 5 5 t 5 b.

Proof: We will indicate the proof for some sample cases and the rest follows in a similar manner. To show (t,.c)+

23 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

$(.,t,.-) and (t,-r) -+ Rt(-,t,.r) continuous with values in EY 2 while t -+ ^R"(*,t,-r) E C (E ) for instance let y E KC Wn and Y ... enclose K in the interior of a compact polydisc KCCn. If a = a a1 (al, . . ., an) write DO" = (a/ayl) . . (a/ayn) and let ... Y .... r = aK be the distinguished boundary of K. We have a Cauchy integral formula (cf. Gunning-Rossi [l])which leads to

Irak! P(<,t,T) - Pk,tO,TO)l ak+l nd

^Rm(*,t,-r) E E for example one wants to show that ARm /At = Y In "m [R (*,t,T) - ?(-,to,T)]/At -+ Rt(-,to,-r) in EY as At = t - to-+

This means that we must prove that Da(Aim/At) -+ Daim(y,to,r)Yt Y uniformly for y E K Rn . This may be written (assuming t > tc for convenience)

Byt by the continuity of 5 we can make ^R"(*,~,T)t 5 E for IE-t0l s(E), uniformly fot "m 1 Y E K, and hence t -+ R (.,t r) E C (Ey). A similar argument

24 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

"m In may be applied to ARt/At - Rtt, etc. QE D

Lema 3.12 For 0 5 T 5 t 5 b the formula (3.22) holds

A pointwise or in E if is replaced by any numerical function Q Y A of argument (y,t,T) satisfying (3.21) with f E Co(E ) where t -+ A 2 A 'A Q(*,t,T) E C (Ev) while initial conditions @(*,T,T) = T(*) E J A A E and Qt(=,~,~)= 0 are stipulated. Such a function $I is neces- Y sarily unique by (3.20).

Proof: Pointwise for (y,t,.r) fixed everything is trivial "m "m since our lemmas concerning properties of R and S make all calculations in (3.18) - (3.20) legitimate for T > 0 and one can let T + 0 in the resulting formulas (cf. Lemnas 3.5, 3.6, 1 "m A 3.8, 3.9 and note that terms -5 S (y,t,c)@(y,€,,T) are continuous h lA since @ satisfies (3.21) with @(Y,~,T) continuous). To work 5 in E we recall that (S,T) + ST : x E -+ E is separately Y 4 Y Y continuous with E a Frechet space and hence this map is con- Y tinuous (see Bourbaki [2] and for vector valued integration see

Bourbaki [3; 41, Carroll [14]). Hence all of the operations in

(3.18) - (3.20) and (3.22) make sense in E OED Y' "m Lemma 3.13 For 0 5 T 5 t I. b we have (t,T) + R (-,t,-c) E "m "m 2 + Co(OM), (t,T) + Rt(*,t,T) E Co(OM), t + R (*,t,T) E C (OM),

Em(*,t,.r) E Co(OM), T + TS;(*,t,r) E Co(OM), and T -+ .111R (-,t,T) E

1 Am 1 C (OM) while, for 0 < T S t 5 b, T + S (*,t,.c) E C (OM).

Proof: Again we will prove this for some sample cases to

25 SINGULAR AND DEGENERATE CAUCHY PROBLEMS indicate a procedure which can then be applied in general. We recall that on bounded sets in 0, the topology of OM coincides with that induced by E and that if t +. $(&,t) E Co(OM)fl C1(E) 1 with t -+ ~)~(-,t)E Co(OH) then t + +(*,t) E C (OM) (cf. Schwartz

[2; 3; 41 and Lemma 3.3). In order to study Da^Rm for example Y we replace by imin (3.4) and differentiate in y to obtain

where the P (y) are polynomials and = "k' The initial Y 1.1 1 conditions are D"im(y,r,~) = Da?(y,~,~) = 0 and by Lemma 3.12 Y Yt the unique solution of (3.47) is given by (3.22) in the form

where the DYkm may be regarded as known functions (by an in- Y duction procedure). Now, referring to Lemmas 3.7 and 3.10, we know Is^m(y,t,S)12 5 c5 and by induction the Dykm(y,E,r) for Y IyI 5 1.1 - 1 will be bounded by polynomials in y independent of (E,T). Consequently all Da?(y,t,r) will be bounded by poly- Y nomials in y depending only on a so that im(-,t,T) is bounded in

OM for 0 5 T S t 5 b and hence from Lemma 3.11 we can say that "m (t,.c) +. R (-,t,.r) E Co(OM). Further if one differentiates (3.48) in t there results

26 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONSOF EPD TYPE for T > 0, with I?$y,t,&)I 5 c7 + c8A(y) for 0 5 T 5 t s by from which follows the desired polynomial bound for D"@ since Yt one can let T + 0 in the resulting inequality for Yt Finally we have from (3.49) for T > 0

"m "m since St(y,t,t) = 1 while Stt is continuous in (y,t,.c) for T > "m 2m+l "m 0. But Stt = -A(y)@ -S and we need only check - tt

Again the resulting estimate on ID"^Rm will hold as -c + 0. Y tt I Hence D"^Rm can be bounded by polynomials in y independent of Y tt "m 2 (t,-c) for 0 5 T 5 t -< b and consequently t + R (*,t,.c) E C (OM).

The rest follows in a similar manner. QED

We can now prove uniqueness (and well posedness) for the solution wm(t) = Rm(t) * T of (3.2) - (3.3) given by Theorem 3.4. As will be seen later there are quicker ways to prove uniqueness when one is working in a Hilbert or Banach space but in order to study the growth of solutions for example it is absolutely

I I necessary to work in "big" spaces such as S or D and for this a more elaborate uniqueness technique is required, such as we have developed. It is sufficient now to deal with the solution

27 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

A p(t,T) = 8(y,t,r)T of (3.4) with initial conditions of the form (3.5) given at t = T. We recall that the map (.S,T) -+ ST : I I "m OM x S -+ S is hypocontinuous and then, given any solution M of (3.4) (with t replaced by C), multiply by ?(y,t,<) as in

A (3.20) to conclude that is necessarily of the form $(y,t,T)T. The same uniqueness argument works for any solution $ of (3.21) I SO that (3.22) holds in S . Hence we have proved

Theorem 3.14 Assume m 2 -1/2 and that A(y) = F(Ax6) satisfies Condition 3.1. Then the unique solution of the equation

2m+l m (3.52) wm + -wt + Axwm = f tt t with f(2) E Co(Sl) and initial data wm(,) = T E SI and w'l](~)= 0

(0 5 T 5 t 5 b) is given by

t (3.53) wm(t,r) = Rm(*,t,T) * T + 1 Sm(-,t,g) * f(g) dg T

Definition 3.15 Following Schwartz [3] we will say that m (3.53) with f(-) E Co(S1), wm(.) = T E S', and w,(T) = 0 is uniformly well posed if the solution (3.53) depends continuously I in S on (t,-c,T,f).

Theorem 3.16 The problem (3.52) with f(*) E Co(Sl), wm(-r) = T E S', and W;(T) = 0 is uniformly well posed in SI with solution (3.53).

Proof: This follows by hypocontinuity. QED

28 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Remark 3.17 The formulas of Lemma 3.3 can be composed with

A I T E S and after an inverse Fourier transformation we have

m t (3.54) wt(t) = -

(3.55) wp) = +w2m m-1 (t) - wm(t)l

These recursion formulas are generalizations of formulas first developed by Weinstein [l; 3; 4; 6; 8; 9; 10; 11; 16; 181 when Ax = -A X’ They were systematically exploited by Carroll [2; 51 in a general existence-uniqueness theory and it turns out that they have a group theoretic significance (see Chapter 2).

1.4 Connection formulas and properties of solutions for EPD I I equations in D and S . Let us return now to the case when Ax = -Ax with A(y) = lyI2 = 1 yi and look at the resolvant im(y,t) given by (3.6). We denote by supp S the support of a distribution S.

“m Lemma 4.1 The resolvant R (y,t) = 2mr(m+l)z-mJm(z),z =

tlyl, of (3.6) is an entire function of y E Cn of exponential

type so that Rm(*,t) E E: and supp Rm(-,t) is contained in a

fixed compact set lykl 5 6 for 0 2 t S b.

Proof: The order p of an entire function g(y) = 1 aaya, y E any is given by (cf. Fuks [l])

29 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

where la/ = 1 ak and if p(g) = 1 with

where llyllC = 1 lyklC then g is said to be of exponential type

5 y. Now looking at @(y,t) = 1 aZkzzk as a function of one

variable z = t(yl we have by Stirling’s formula (cf. Titchmarsh “m [l]) that p = 1. This means that IR (y,t)l 5 6 exp(B+E)lzlC for

any E > 0 where B is the type of imas a function of z. Since 2 1/2 lYlC = Yk) IC (1 lYk1E)1/2 5 ZlYklC = llyllc One has then

I Bt 5 Bb = B

The conclusion of the Lemma now follows from the Paley-Wiener- Schwartz theorem see Schwartz [l3) . QE D

I Theorem 4.2 If Ax = -Ax, m 2 -1/2 0 5 t 5 b, and T E D

then wm(t) = Rm(-,t) * T is a solution of (3.2) - (3.3) in D’ I depending continuously in D on (t,T).

Proof: By Lemma 4.1 supp Rm(-,t) is contained in a fixed

compact set for 0 2 t 5 b so that Rm(.,t) * T makes sense for

T E DI. Furthermore the set {Rm(*,t); 0 i t 6 b} is bounded in I E (cf. Ehrenpreis [3], Schwartz [l]) and one knows by Schwartz

[l]that the map (S,T) -+ S * T : El x DI -f D’ is hypocontinuous.

To check the differentiability of t -+ Rm(.,t) in El we recall

30 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

I that on bounded sets in E the topology coincides with that in- I duced by Oc (cf. Schwartz [l],p. 274, where the Fourier trans-

I form is shown to establish a topological isomorphism between E and Exp 0M and note that on sets BC Exp OM of bounded exponential type the topology is that induced by OM). The result follows immediately. QED

For uniqueness in Theorem 4.2 we can go to a direct expression for "R"(y,t,r) derived in Carroll [2; 51 and then follow the procedure involving (3.20). Thus one can write for m 2 -1/2 not an integer

where t = tm, z1 = TM,and y, = r(m+l)r(-m)/Z, while for m 2 -1/2 an integer we have

where Nm denotes the Neumann function (cf. Courant-Hi1 bert [Z],

Watson [l]).Some analysis of the Bessel and Neumann functions, which we omit here (see Carroll [2; 51 for details), leads to

2 ^m Lema 4.3 Wnen A(y) = IyI the resolvants R (y,t,T) are

31 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

entire functions of bounded exponential type for 0 5 T 5 t 5 b with {Rm(*,t,=)} a bounded set in El. Similarly {R:(*,t,S)}C El

is bounded for 0 5 6 5 t 5 b.

Using for example (3.39) we can say that given A(y) = IyIL,

{S"(*,t,S)l C El is bounded for 0 5 6 5 t 5 b and certainly

Lemmas 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, and 3.12 remain valid. I Lemma 3.13 holds with OM replaced by Exp OM = FE c OM and if we take inverse Fourier transforms in (3.18) - (3.20) the result- I ing convolution formulas will be valid for T E D . Hence we have

Theorem 4.4 The solution wm(t) = Rm(-,t) * T E DI of (3.2)- I (3.3) when Ax = -Ax and T E D , given by Theorem 4.2, is unique.

Similarly the problem (3.52) with f(*)E Co(D1), w"(T) = T E D',

I and W:(T) = 0 is uniformly well posed in D with solution given

by (3.53). In Going back now to the resolvant R (y,t) in OM or Exp OM we

exploit the Sonine integra formula (cf. Rosenbloom [l],Watson

[l])to obtain for m > p Z -1/2

We note in passing that for m-p integral (4.6) is also a direct

consequence of the recursion relation (3.12) (cf. Carroll-Silver

[15; 16; 171, Silver [l])and this will be utilized in Chapter 2

where an explicit derivation is given; analogous formulas in

32 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE other group situations will also be derived. We recall next the formula (cf. Watson [l])

for p 2 0 an integer. This leads to the formula

provided m f -1, -2, . . ., -p. We observe that (4.8) gives resolvants for va ues of m < -1/2 and it should be noted here that

rn -2m+l I jm(z) = z J-,(z satisfies jm + (T)jm.+ jm= 0 so that (cf.

A (3.6)) R-m(y,t) = 2-mr(l-m)zmJ-m(z), for z = tm and any

A -2m+l A nonintegral m 2 0, satisfies the equation RiF + (7)Rtm + h A A(y)"Rm = 0 with initial data R-m(y,o) = 1 and R-m(y,o) = 0. Now t from (4.8) with nonintegral m < -1/2 we can choose an integer p so that m + p 2 -1/2 and express InR in terms of derivatives of

A RmP, whose properties are known from Section 1.3. In particular

(4.8) holds in E and also in OM or Exp OM(cf. Lemma 3.13 and the Y remarks before Theorem 4.4) so we can state, referring to (3.11),

Am Theorem 4.5 For nonintegral m < -1/2 resolvants R (y,t) of the form (3.6) obey (4.8) and thus inherit the properties "m of the R for m 2 -1/2. Also (4.6) holds in OM or Exp Or,,.

Taking inverse Fourier transforms in (4.6) and (4.8) and

I I composing the result with T E S or T E. D (when Ax = -Ax) we obtain

33 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Theorem 4.6 For m > p 2 -1/2 the (.unique) solution of (3.2)- I I (3.3) in S (or D ) for index m is given by

where wp satisfies (3.2) - (3.3) in SI (or 0') for index p. Similarly if wm+P is the (unique) solution of (3.2) - (3.3) in I I S (or D ) for index m + p 2 -1/2 (p an integer) then

I I is a solution of (3.2) - (3.3) in S (or D ) for index m provided m + -1, . . ., -p.

Remark 4.7 Formulas of the form (4.9) - (4.10) were first discovered by Weinstein (loc. cit.), in a classical setting,

using different methods. In particular Weinstein observed that

if w-~satisfies (3.2) with index -m then t -+ w"(t) = t-2"CI-m(t) satisfies (3.2) with index m. This fact, plus a version of (3.54) , leads to (4.10) for example. Weinstein's version of

(4.9) involves p = n - 1 and is obtained by a generalized method of descent (see e.g., Weinstein [3]). We note also here that

there is no uniqueness theorem for solutions of (3.2) - (3.3) 2s s when m < -1/2 since ifm =-s (s>1/2) then t -f um(t) = t w (t) satisfies (3.2) for index m with um(o) = uF(o) = 0; here ws is the solution of (3.2) - (3.3) for index s.

Remark 4.8 The exceptional cases m = -1, -2, . . . have

34 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE been treated by Blum [l;23, Diaz-Weinberger [Z], Diaz-Martin

[lo], B. Friedman [l],Martin [l],and Weinstein (.loc. cit.) and we refer to these papers for details. Here for m = -p with p 2

1 an integer we may take

h (cf. B. Friedman [l]). This resolvant R-p satisfies (3.4) -

h (3.5) for index m = -p with T = 1 and one may write

m m.." 2k (4.12) zpNp(z) = 1 ckzZk + zZp log z 1 ckz k=O k=O Consequently the 2pth derivative of ^R'p in t will have a logarithmic singularity at t = 0. However, writing formally

m (4.13) w-P(t) = R-P(-,t) * T = 1 cktZk(AxS * T) k=O

we see that the logarithmic term varnishes if A!T = 0, in which event w-~will depend smoothly on t.

Remark 4.9 We note that if one takes p = 1 in (4.8), with m replaced by m - 1, the result is exactly (3.12).

We will now discuss some growth and convexity theorems for solutions of (3.2) - (3.3) when Ax = -Ax. Such results were first discovered by Weinstein [9; 10; 151 in a classical setting and subsequently generalized and improved in certain directions by Carroll [l;2; 51 in a distribution framework; we will follow

35 SINGULAR AND DEGENERATE CAUCHY PROBLEMS the latter approach. (Other kinds of growth and convexity theorems for singular Cauchy problems were developed by Carroll

[18; 19; 251 and will be treated later.) Referring to (4.6), the basic fact one requires for these theorems is that

Rp(*,n) 2 0 for some p 2 -1/2 (i.e., Rp(-,n) should be a positive measure), in which case all Rm(-,t) 2 0 for m 2 p. For general

Ax not too much is known in this direction but when Ax = -Ax it follows from Section 2 that Rn/'-'(=,n) = ux(n) 2 0. Another 2 case of interest involves the metaharmonic operator Ax = -A a X - m n n where R (-,n) L 0 for m > 7 - 1. Indeed for m > 7 - 1 and r = (1~:)"~ 5 t one has (see Carroll [5] and cf. Ossicini [l])

2 2 m-n/2 m - r(m+l)(t -r ) 1 2k [a2$-r2) Ik20 IT n/2 t2m k=O 2 k! r(m-$+k+l) while p(*,t)= 0 for Irl 2 t; in particular then Rm(*,t) E El.

Now, general theorems of growth and convexity involving as- k sumptions of the form (-Ax) T ;1 0 were developed in Carroll [2;

51 using the notions of value and section of a distribution introduced by Lojasiewicz [l]. This can be simplified somewhat in dealing with Ax = -A since AT 2 0 implies that T is an almost X subharmonic function. (see Schwartz [l])and hence T automatically has a "value" almost everywhere (a.e.). We recall briefly some notations and resu ts of Lojas ewicz [l].

36 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Definition 4.10 A distribution T is said to have the value + c at x if lim Tx +xx = c as X -+ 0 (i.e., I positive measure the notion of value coincides with that of den-

sity. One may fix x in a distribution T if lim Tx = 0 x,t I + 0 St E D as X -+ 0 and St is said to be the section T of T. xo ,t

Thus one must have -+ when $I E Dx, S,taAn1 $ E Dt, and I$(x)dx = 1. Remark 4.11 It is proved in Lojasiewicz [l]that if 1 X,t is a measure then a.e. in x we may fix x = xo and the section

I is a measure in D Further if xo may be fixed for T lxo, t t' x,t then xo may be fixed for (a/at)T and (a/at)Tx,t Ix ,t = X ,t 0 a/atTx ,t' 0 For the following theorems see Carroll [5].

Theorem 4.12 Let w:(t) = Rm(*,t) * T E D1 be the solution X of (3.2) - (3.3) for Ax = -Ax with m 2 $ - 1 and T E D' satis-

fying AT 2 0. Then a.e. in x the section t -+ w! (t) is a non- A 0 decreasing function of t with wm (t) >Tx' xO 0 n Proof: Let m > 1 and set p ---- 1 in (4.6) to obtain n 2 - 2

Using (3.11) (transformed) and composing with T we have (cf.

(2.5) where Rn"(*,t) = Ax(t))

37 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Now we may consider wm(t) as a distribution wm in two variables x,t I1 I I (since Dt(Dx) = D algebraically and topologically with Et(Dx)

I. x,t C Di(D;) - cf. Schwartz [5]) and here one may extend ux(t) I and Ax(t) as even functions of t for t < 0. In Dx one has t wm(t) - T = I wT(r)dr which, by (4.16) with AT 2 0, is a positive 0 ." I measure for t L 0. Writing T = T x It in D it follows that Y X ,t m w - T will be a positive measure in D for t L 0 and hence X,t x,t m - we can fix x = x a.e. in x for w and T together (cf. Remark 0 x,t 4.11 and recall that T is an almost subharmonic function). For m such xo, (a/at)wx ,t = (~y)~~,~by Remark 4.11, and, by (4.6), 0 m (W") > 0 for t 2 0. Hence t + w; ,t = w (t) will be a t X0,t - 0 xO nondecreasing function for t 2 0 of bounded variation (cf.

Schwartz [l]).We conclude that w: ,t Z Txo for t 2 0 and to ro spell this out let (p E Dx, (p z 0, Ji(x)dx = 1 and set $,(E) = (cf. Definition 4.10). Then from (4.16) the contin-

uous function t + = is nondecreasing for C,t' A t 2 0. Consequently for t > 0 fixed + C,t' A and letting A + 0 we obtain wm 2 Tx . The result for m = X0,t 0 - 1 is now immediate from (2.5) and the remarks above. QED

Let us now combine (4.8) with Lemma 3.3 when A, = -Ax in

order to obtain a formula for Rm(-,t) (m 2 -1/2) in terns of

38 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Rm+P(*,t) and Rm+p+l(-,t), where the integer p is chosen so that m + p 2 2 - 1. Some simple but tedious calculations yield (see Carroll [2; 51 for details)

Lemma 4.13 For -1/2 5 m < n/2 - 1 the resolvant R"(*,t) for

(3.2) - (3.3) (i.e., T = 6 in (3.3)) given by F-l applied to (4.8) can be written

q-1 (4.17) Rm(-,t) = ( 1 ak(Fl)t2k(A6)k) * Rm+P(-,t) k=O

q- 1 + ( 1 Bk(n)t2k(A6)k) * Rm+P+l(*,t) k=O n where p is an integer chosen so that m + p 2 7 - 1, q = 1 + [q]([u] denotes the largest integer in u), ak(m) 2 0,

Bk(m) 2 0, and ~1 (m) = 1. 0 n Theorem 4.14 For - 1/2 5 m < 7 - 1 let the integer p be n chosen so that m + p 2 7 - 1. Let q = 1 + [q]and assume k T E D' satisfies (A) T 2 0 for 1 F k 5 q. If wm(t) denotes the

solution of (3.2) - (3.3) for Ax = -A(t 2 0) then a.e. in x the rn section t + wx (t) is a nondecreasing function of t with 0 wm (t) 2 T for t -> 0. xO xO

Proof: We convolute (4.17) with T to get wm(t) and dif- m+p+l ferentiate, using (3.11). Since Rm+P(*,t), R (*,t), and p+p+2 (-,t) are positive measures we have again that wT(t) -> 0

in Dx for t 2 0. Hence, as in Theorem 4.12, we may fix x = xo m I m a.e. for w E D and t + w = w (t) will be a x,t x,t X0't xo

39 SINGULAR AND DEGENERATE CAUCHY PROBEMS nondecreasing function of bounded variation (t 0). Further wm(t) = Rm+P(.,t) * T + l(*,t) where l(.,t) _> 0 in Di for t L 0

(we recall here that ao(m) = 1). From Theorem 4.12 we know that

Rm+'(=,t) * T = wm+P(t) satisfies wm+P(t) 2 T a.e. in x when xO xO m AT 2 0 (which holds since q L 2). Hence, a.e. in x, w (t) L xO w!+P(t) L Tx . QED 0 0 n Corollary 4.15 For -1/2 5 m < 7 - 1 choose p as in k Theorem 4.14 and assume (A) T 5 0 for 1 5 k 5 q - 1. Given w"(t) as in Theorem 4.14 then a.e. in x the section t + wm (t) satis- xO fies wm (t) 2 T for t >- 0. xO xO Proof: We note first that p L 1 so that q - 1 = [q]L 1. Then from (4.17) one writes again wm(t) = Rm+P(-,t) * T + l(*,t) I where l(*,t) 2 0 in Dx for t 2 0 under our hypotheses. As in the proof of Theorem 4.14 it follows that wm (t) L T a.e. in x. xO xO

Let us write now Lm = 2n + then as noted a /at + -t a/at; by Weinstein, for m f 0, Li = (2m)2t-2(2m+1) a2/aa(t-2m)y while 222 for m = 0, L: = t- a /a (log t). Referring now to (3.2) with

A = -A we have Lkwm = awm and using (4.15), composed with T E X I D , for m > n/2 - 1 it follows that

40 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE since Lt = C2LCt. Thus if AT 2 0 then Lh wm(t) is a posi- -- 1 tive measure for t t 0. Similarly from f4.17) with - 1/2 5 m < E 2 - 1 we convolute with T D: to obtain

q-1 2k m+p+l + 1 Bk(m)t (R (-,t) * A~+’T) k=1 k Hence ifA T 2 0 for 1 5 k 5 q then Liwm(t) is a positive mea-

l sure in Dx for t t 0.

Theorem 4.16 Let wm(t) denote the solution of (3.2) - I n (3.3) for Ax = -A and let T E D satisfy AT 2 0 for m 2 - 1 n k while, for - 1/2 5 m < - 1, A T 2 0 for 1 i k 5 q = 1 + n [q]where the integer p is chosen so that m + p t - 1.

Then for t > 0, a.e. in x, the section t -+ wm (t) is a convex 2-n n xO n function of t for m ,t 2 - I, of t-2mfor -1/2 <, m < 7 - 1 (m f 0), and of log t for m = 0.

Proof: By (4.18) - (4.19) we know that for t > 0, 2 Zm I (a /a-c )w (t) L 0 in Dx under the hypotheses given, where T = n n t2+ for m 2 2 - I, -c = t-2mfor -1/2 I m < 7 - 1 (m t 01, and T = log t for m = 0. Now by Theorems 4.12 and 4.14 our hypotheses insure that x = xo may be fixed a.e. in x for m w (t 2 0) and by Definition 4.10 with Remark 4.11 it follows x,t that a2wm Z 0 in Difor t > 0. Hence by Schwartz [l] xo ,t /a? t -+ wl (t) is a convex function of t. QED 0

41 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Remark 4.17 Of particular interest in the EPD frameword of course is the wave equation when m = -1/2 and we see that in the n n-1 hypotheses of Corollary 4.15 -1/2 + p 2 2 - 1 means p 2 -. 2 Note here that in dimension n = 1, -1/2 5 m < -1/2 is impossible so that we are in the situation of Theorem 4.12. Now q - 1 = [q]so that, given n even, p = n/2, while, for n odd (n t 3), n-1 n +2 p = 2; thus for n even q 1 = [-1 while for n odd q 1 = - 4 - [!$!-I.These results on the number of positive Laplacians of T sufficient for a minimum (or maximum) principle improve

Weinstein's estimates and are comparable with results of D. Sather [l;2; 31. Sather in [3] for example sets N = 9for n-3 N+2 n even and N = for n odd with a = = for n even -2 [-I2 [?I N+l n while a= [%'Ifornodd; similarly he sets b = [T] = [,I for n- 1 k n even while b = for n odd. Then he requires that A T t 0 [-I4 for 1 5 k 2 a and Akw;'/2(o) 2 0 for 0 5 k 5 b in order that w-l/'(x,t) 2 T(x) for t 2 0. Thus our results seem at least equivalent and certainly more concise than those of Sather. For further information on maximum and minimum principles for Cauchy type problems we refer to Protter-Weinberger [l]and the bibliography there; cf. also Agmon-Nirenberg-Protter [l],Bers [l],

Protter [7], Weinberger [l]. For growth and convexity properties of solutions of parabolic equations with subharmonic initial data see Pucci-Weinstein [l].

Remark 4.18 The existence of positive elementary solutions

(or resolvants) for second order hyperbolic equations is in

42 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE general somewhat nontrivial and we mention in this regard Duff

[l;21, in addition to Carroll [2], where the problem is discussed.

1.5 Spectral techniques and energy methods in Hilbert space. F rst we w 11 present some typical theorems of Lions for

EPD type equations, using "energy" methods, which were discussed in Carroll [8]. Lions proved these results in 1958 in lectures at the University of Maryland (unpublished). There- after follow some theorems in Hilbert space based on a spectral technique developed by Carroll [4; 6; 8; 10; 111 (suggested by

Lions as a variation on Carroll's Fourier technique). The two approaches (energy and spectral) are not directly connected and lead to weak and strong solutions respectively.

Thus, regarding energy methods (cf. Carroll [14] and Lions

[5] for a more complete exposition), let (u,v) -+ a(t,u,v) :

V x V +. B be a family of continuous sesquilinear forms, Vc H, where V and H are Hilbert spaces with V dense in H having a finer topology. One can define a linear operator A(t) such that a(t,u,v) = (A(t)u,v),, for u E D(A(t))CV and VE V (i.e., u E

D(A(t)) if V +. a(t,u,v) : V -+ B is continuous in the topology of 1 H). Assume t +. a(t,u,v) E. C [o,b] with a(t,u,v) = a(t,v,u) and consider

Problem 5.1 Find u(-) E C2(H) on [o,b], u(t) E D(A(t)), t +. A(t)u(t) Co(H) on [o,b], and

43 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

where k E a, q(-) E Co(o,b] is real valued with q(t) -+ 03 as t -+ 0, and f E Co(H) on [o,b]. Note here that if u(o) = T E

D(A(t)) for t E [o,b] then vit) = u(t)- T satisfies (5.1) with f replaced by f(*) - A(-)T (assuming the latter is continuous on [o,b] with values in H - which holds if A(t) = A). This can be transformed into a weak problem (which is more d -' tractable) as follows (== ).

Problem 5.2 Find u E L2(V) on [o,b], u(o) = 0, ulfi E 2 L (H) on [o,b], such that given f/GE: L2(H)

I1 (5.2) jbla(t,u,h) +kq(u',h)" - (u ,h )H - (f,h)Hldt = 0 0 for all h E L2(V) with hJii E L2(H), hl/J;i E L2(H), and h(b) = 0.

1 Theorem 5.3 Let t -+ a(t,u,v) E C [o,b], a(t,u,v) = a(t,v,u) q E Co(o,bl, q > 0, q(t) -+ 03 as t + 0, Re k > 0, and a(t,u,u) 2 2 clllullv for a > 0. Then there exists a solution of Problem 5.2.

Proof: Set h = e-Yt+l for y real (to be determined) and

b (5.3) Ey(u,+) = I {a(t,u,e-Yt+l) + kq(t)(ul ,e-Yt+l)H 0

1- - (u ,(e Yt+l)l)Hldt

2 Let F be the space of functions u E L2(V), ulG E L (H), 2 '2 u(o) = 0, while G is the space of 4 E L (V), + E L (V), I 2 I1 2 I 9 4 E L (H), 9 /4E L (H), $(o) = 0, and + (b) = 0. Clearly

44 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

G CF and we put on G the induced topology of F defined by ,f)( IIull; + qllu'l!i)dt = IIuIIF2 so that G is a prehilbert space. 0 b- Then if u E F satisfies E (u,$) = / (f,e Yt$')Hdt for all $ E Y 0 G it follows that u is a solution of Problem 5.2. Now Ey(u,$) is a sesquilinear form on F x G with u -f E (u,$) : F.+ 4 con- yt tinuous for 4 E G fixed. Further, a(t,$,e yt$) = at(t,$,e-Yt$) + 2a(t,$,e-Yt$') - ya(t,$,e-Yt$) so that,

(5.4) 2Re I:a(t,$,e-Yt$')dt = a(b,$(b),$(b))e-yb b - I at(t,$,$)e-Ytdt + Y 0

bi (5.5) -2Re ($ ,(e-Yt$')l)Hdt 0 b = I [-e-yt &11$11: + 2ye-ytII$'lli]dt 0 b '2 = 11$'(0)11 + e-ytII $ I] H dt ' 0

Hence, for y > c/a, IEy($,$)I 2 811$11; (note the second term in 2Re E ($&)) and by the Lions projection theorem (cf. Y Carroll [14], Lions [5], Mazumdar [l;2; 3; 41) there exists

E (f,e-Yt$')Hdt, since $ -f u F satisfying EY (u,$) = f(f,e-yt$')Hdt is a continuous semilinear form on G. QED 0

Remark 5.4 For an interesting generalization of the Lions

45 SINGULAR AND DEGENERATE CAUCHY PROBLEMS projection theorem and applications see Mazumdar [l;2; 3; 41.

We note that in Problem 5.2 q could become infinite at other points on [o,b] provided the "weight" functions h are suitably chosen; if q > 0 but q the problem can be solved directly as in Lions [5].

1 Theorem 5.5 Let t + a(t,u,v) E C [o,b], a(t,u,v) = 1 a(t,v,u), q E C (o,b],q > 0, q(t) +- as t + 0, Re k > 0, and 2 a(t,u,u) 2 allullv for a > 0. Then a solution of Problem 5.2 is unique.

I Proof: Let h be given by h(t) = 0 for t 2 s with h - kqh

= u on [o,s); then some calculation shows that h is admissable z in Problem 5.2 and, as E + 0, lim a(E,h(E),h(E)) = 8 2 0.

Taking real parts in (5.2), with f = 0, one obtains with this h,

+ jo at(t,h.h)dt = 0

I I where a(t,h,h) = 2Re a(t,h,h ) + at(t,h,h). Consequently

S 2 (5.7) I (2aq Re k - c)llhllVdt 5 0 0

and since q(t) -+ (4 as t + 0 one may choose so small enough so that 2aq Re k - c > 0 in [o,so] which will imply that h = 0 and u = 0 in [o,so]. In this step only q E Co(o.so] is needed; to extend the result u = 0 to [so,b] one can resort to standard methods (cf. Lions [5]) where q E C'[s,,bl is used. QED

46 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Remark 5.6 There are also versions of Theorems 5.3 and 5.5 2 2 due to Lions when a(t,v,v) + Al1vllH 2 cill-vllv (cf. Carroll [81) but we will omit the details here. We note that no restrictions have been placed on the growth of q(t) as t * 0; however the solution given by Theorem 5.3 is a weak solution and not necessarily a solution of (5.1).

We consider now the operator

for 0 5 T 5 t 5 b < my where ci E Co(o,b] with I: Re a(<)d< -+to as T -+ 0, B E Co[o,b], and y E Co[o,b]. Let A be a self adjoint (densely defined) operator in a separable Hilbert space H with

(Au,u)~ + 61Iulli 2 0 and consider for Q E Co[o,b]

2 Problem 5.7 Find w E C (H) on [~,b] (0 5 T -5 t S b < m), w(t) E D(A), and t -+ Aw(t) E Co(H) on [~,b], such that

(5.9) (L + Q(t)A)w = 0; W(T) = T E H; Wt(~) = 0

It is to be noted that one may always assume A 1 0 since a - .v change of variables A = A + 6 and y = y - SQ can be made. Now from the von Neumann spectral decomposition theorem (cf. Dixmier [l], Carroll [14]) it follows that there is a measure v, a v-

measurable family of Hilbert sapces X -+ H(X) (cf. Chapter 3), and an isometric isomorphism 0 : H -+ ff = I"H(A)dv(A), such that A is transformed into the diagonalizable operator of multiplication

47 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

IIBT(A)l[H(A)) and T E D(A)-if and only if in addition [ A2118Tll$A)dv < - where BT is the image of T E H in H; we write here BT for the family (BT)(A) determined up to a set of v-measure zero. For various other applications to differential equations see e.g., Berezanskij [2], Brauer [l],Girding [l],

Lions [6], Maurin [l;2; 31. Now by use of the map 8 (5.9) is transformed into the equivalent problem

(5.10) (L + AQ)Bw = 0; ew(-c) = BT E H; Bwt(-r) = 0

with t -+ Bw(t) E C2(H) and t -+ ABw(t) E Co(H) on [o,b] (note that

BAw = BAB-'Bw = ABw). This essentially reduces Problem 5.7 to a numerical problem, as in the case of the Fourier transform, and we will construct suitable "resolvants" Z(X,t,-c) and Y(A,t,-c) ^m %l analogous to the R (y,t,-c) and S (y,t,-c) respectively of Section 3 so that for example

(5.11) (L + AQ)Z = 0; Z(A,T,T) = 1; Zt(A,-c,-r) = 0 in which event Bw = Z(A,t,-r)@T will be a solution of (5.10).

Many of the techniques of Section 3 will be employed and this will enable us to shorten somewhat the exposition here.

Let us assume Re a ?: 0 for 0 < t <_ s and consider P(t) = b exP (-1 (a(<)+ B(S))dS) (cf. (3.23)). Then P(t) -+ 0 as t -+ 0 t and

48 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

for 0 5 5 5 u 5 b. (In the terminology of Feller [2] t = 0 is an entrance boundary.) As in (3.23) - (3.24) we obtain from

(5.11) (PZt)t + P(y+AQ)Z = 0 and

rt 1 1'5 (5.13) Z(A,t,T) = 1 - 1, rn 1, P (5)[Y (51 +AQ (51 IZ( A ,5 y~ 1 which we write again in the form Z = 1 - J Z. Setting F(t,r) =

1: 1: /llP(5)/P(o)ldgdo (cf. (3.27)) we have F(t,-c) < 03 with h h F(t,-r) 5 F(t,o) = F(t) 5 F(t) S F(b) for t 5 t 5 b. The solution m .. of (5.13) is given again formally by Z(A,t,T) = 1 (-l)kJk1 k =n (cf. (3.25)) and one may proceed as in Section 113-to prove

m Lemma 5.8 Given Re CL 2 0 on (o,s] the series 1 (-l)kJk 1 k=O converges absolutely and uniformly on (0 i T tiblx

$ compact (A E r), and represents a continuous function of (A,t,T), analytic in A, which satisfies (5. 3). Sim

Zt(A,t,-c) is continuous in (A,t,.c) and analytic in A.

Proof: The statements for Zt(A,t,-r) follow upon differentiating (5.13) in t (cf. Lemma 3.5). We note also that if

Section 3. QED

NOW, as in (3.30), we must examine the behavior of a(t)Zt(A,t,o) as t -+ 0. Thus (cf. (3.31)) we set

49 SINGULAR AND DEGENERATE CAUCHY PROBLEMS a(t) jt(P(<)/P(t))df = H(t,T) with H(t) = H(t,o). Let us assume T 1 '2 now that a E C (o,b] with 0 = a /a E Co[o,sl, la1 2 kRe a on

[o,sl, and a(t)/a(S)I 5 N for 0 5 5 5 t 5 s. If lim H(t) =

H(o) exists as t -+ 0 then clearly (cf. (5.13) upon differentiation) lim a t) Z (X,t,o) = -H(o)[y(o) + AQ(o)]. Now consider tt A A A (-jga(n) dn) d5 = H(t,T) with H(t) = H(t,o). Since JT A B E Co[o,bl it is easily seen that IH(t) - H(t)l can be made arbitrarily small for t sufficiently small. Indeed if B(t,S) = rt exp (-1 B(q) dn) ther;, by continuity, given E there exists 6(~) 5 such that lB(t,5) - 1 I 5 E for t 5 ti(&). Hence for t 5 min(s,b(&)) one has t -15Rea(n)dnt (5.14) lG(t)-H(t)l 5 la(t)l 1 11 - B(t,S)l e dS 0

t -lSRea(nt )dn 5 EkN Rea(5)e dS = skN 0

A A To find now H(o) = lim H(t) as t -+ 0 we integrate the definition

A of H(t,T) by parts to obtain

Letting T -+ 0 we have

50 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPDTYPE

t h t where f(t,S) = a(t) exp (-/cr(o)dn). But H(.t) = i f(t,S)dS t 0 so (5.16) can be written as h(t) 5 1 = f(t,<)(l-@(S))dS. Evi-

A 10 dently h(*) is continuous as t -+ 0 and choosing t smal enough

4) 1

Hence (1-@(o))H(t) -+ 1 as t -+ 0 and if @(o) f 1 this means

h lim H(t) = lim H(t) = H(o) = l/(l-@(o)) as t -+ 0 from which follows that t -+ a(t)Zt(A,t,o) is continuous in t as t -+ 0 with lim a(t)Zt(A,t,o) = w.It may then also be shown easily that a(t)Zt(A,t,r) is continuous in (A,t) for T >_ 0 fixed and analytic in A for (t,r) fixed (0 5 T 5 t 5 b); similar properties are then transported to Ztt(A,t,-c) by (5.11).

Consequently

1 '2 0 Lemma 5.9 Let ct E C (o,b] with 6 = a /a E C [o,s], la1 2 kRea on [o,s], and la(t)/a(<)l 5 N for 0 5 < 5 t 5 s. 2 Then Z(A,-,T) E C [o,bl and satisfies (5.11) for 0 5 5 2 t 5 b while ct(t)Zt(X,t,r) and Ztt(A,t,r) are continuous in (A,t) and analytic in A.

Proof: It remains to prove that @(o) 1 and we suppose

the contrary. Then Re @ -+ 1 and Im @ -+ 0 and we can make 1 1- IRe@(t) - 11 5 E for t 5 B(E). Now - -

51 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

t -i $(S)dS (z < t) with lim = 0 as T + 0 since Rea(-c) + m. T rt Hence l/a(t) = @(S)dSand for t 5 rnin(s,d(E)) and E < 1 we -10 can write

which violates the fact that Rea(t) > 0 for t small. QED

We can now derive some bounds for the resolvant Z as in Section 3 (cf. (3.32) - (3.33) and Lemna 3.7). Here one multiplies (PZ5)5 + P(y+hQ)Z = 0 by P-'(g)T5(h,5,~) and integrates from T to t to obtain

t + (y+hQ)ZF5dS = 0 T 1 Assume now that Q E C [o,b] with Q real and 0 < q 5 Q(t). Adding (5.19) to its complex conjugate and noting that d d 2 2Re(Z 7 ) = - IZ 1' with 2Re = - IZ[ we have 5 55 d5 5 Zz 5 d5

(5.20) IZt(h,t,e)12 + 2

It is convenient now to write a + B = a + a a1 = a on 12where ."- Y - (o,s], s < s, a1 = 0 on [s,b], and a1 = a[(~-t)/(s-s)]on [s,s].

Then a1 E Co(o,b] with :[all 5 klReal on (o,s] and lal(t)/al(5)l 5

52 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

N~ for 0 5 5 5 t 5 b. Then from (5.20) we obtain

I d5

+ IZ 1') and set 5 = 2 = = SUP c 1 sup IRea21, c2 sup Iyl, and c3 IQ'I (on Co,bl). It may be assumed that A 2 l/q since this can always be assured by adding if necessary a constant multiple of Q to y (e.g., set A' = A + 1/q aed yl = y - ($)a so that yl + X'Q = y + AQ with A' 2 l/q). Then from (5.21) we have

+ (c2+c

+ c4JTI

= + where c 4 c1 c2. Now let us state a version of Gronwall's lena which will be useful (see Carroll [14] or Sansone-Conti [l] for proof).

1 Lemma 5.10 Let u E Co, R E L with R L 0, and + absolutely t continuous. Assume u(t) 5 +(t)+ R(o)u(q)dn. Then t

53 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Applying Lemma 5.10 to (5.22) with R = c4, t = T~ u(t) = IZtI2 + XQ(t)lzl 2 , and $(t) = XQ(.r) + (IC3+c2)j:l;12dS it fol-

lows that

(5.24) IZt12 + XQ(t)lZI2 <_ hQ(.r) exp c4(t--r)

where c5 = exp c4b. In particular

Since X 2 l/q we have -1 5 1 and using Lemma 5.10 again there Aq results from (5.25)

c5Q(-c) (5.26) 1ZI2 < -exp c6b = c -q 7 c5 where c6 = -(c +c ). Putting this in (5.24) we obtain q32

(5.27) lZtlL 5 c8X + c 9

1 Lemma 5.11 Assume Q E C [o,b] with Q real and O

(5.26) - (5.27) for 0 5 T 5 t 5 b while la(t)Ztl <_ clOA + cll.

Proof: There remains only the last statement. Let c12 =

sup IQI on [o,b], c13 = sup la21 on [o,b] (recall that a + 6 =

54 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE al + a2), and c14 = sup exp (- I: Re a2(q)drl) for 0 5 5 5 t 5 b. Then differentiate (5.13) in t and for t 5 s we have (cf.

(5.12))

c + C16 5 15A where (5.26) has been used. QED

Remark 5.12 Under certain mild additional hypotheses the

assumption $I E Co[o,s] is sufficient to insure that

la(t)/a(T)l 5 N for 0 5 T 5 t 5 s (cf. Lemmas 5.9 and 5.11).

In particular this holds if $(o) f 0 or if $//@IE Co[o,s] for .., .., some s 5 s (see Carroll [8] for details).

We construct an "associate" resolvant Y(A,t,T) , corres- "m ponding to S (y,t,-c) in Section 3, by the same technique to obtain (cf. (3.16) - (3.17))

Then given Rea 2 0 on (o,s], (5.12) holds and one can differen-

tiate (5.13) in T to obtain Y(X,t,-c) = U(t,.r) - (J Y)(A,t,-r)

where U is given as in (3.34) by U(t,.r) = (P(T)/P(a))do =

55 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

tI(u,T)du and J is defined by (5.13). The formal solution is 'T 00 again Y = 1 (-l)kJk U (cf. (3.36)) and to show that U(t,T) k=O is continuous in (t,T) we need only consider the critical point

T = 0. Thus defining evidently U(t,o) = 0 for t 2 0 (note that our assumptions preclude a case where a(t) = 0 as when m = -1/2

in Section 3) one has lI(u,~)l5 M for 0 5 T 5 u 5 b (cf. (5.12)) .., .., and we write U(t,-c) = I(u,.r)do where I(u,T) = 0 for u < T. 1: 1: w Now let (t,T) -+ (to,o) for to -> 0; then I(o,T) -+ 0 for any u 9 0 .., .., as T -+ 0 with 1 I(o,T) 1 5 M and we may invoke the Lebesque bounded convergence theorem (cf. Bourbaki [3]) to conclude that

U(t,T) -+ 0. Hence the terms Jk U will be continuous in (A,t,T) and analytic in A with lU(t,-c)l 5 Mb. By previous analysis (cf.

Lemmas 3.8 and 5.8) we have

Lemma 5.13 Given Recl L 0 on (o,s] the series Y = 00 1 (-l)kJk U converges absolutely and uniformly on (0 5 T 5 k=O t 5 bl x r, rc F compact (A E r), and represents a continuous

function Y(A,t,-c) of (A,t,-c), analytic in A, which satisfies Y = U - J Y (J given as in (5.13)). One has (5.29) and for T > 0

(5.30) holds. Moreover Y(A,T,T) = 0, Y(A,t,o) = 0, and, for

T > 0, Yt(A,T,T) = 1.

Proof: This goes as in the proof of Lemma 3.8 (note here

that Ut = P(T)/P(t), UT = -1 + (a+B)U, and from (5.301, cf. rt

56 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Now YT is defined by (5.30) and does not necessarily tend to infinity as T -+ 0 (cf. (3.42)). We need only examine a(T)Y(A,t,T), or a,(-c)Y(A,t,T), as T -+ 0. As in (3.40) m1 al(T)Y(h,t,T) = 1 (-l)kJk CX~(T)U(*,T)so we look first at k=O

I aB2( 0, where B2(o,~) = and $1 = a;/a: = $ for sufficiently a0 w o small arguments (e.g., t 5 s); here B2(o,~) = exp (-i aZ(rl)do) T is Cm in (a,~)with IB2(o,~)I 5 c14 (cf. Lemma 5.11). We examine first the term A(t;-c) = al(q)dn) as T + 0 and one

2m + can compare here with Section 3 where a(t) = 7 and

A(t,.r) = (T/t)2m -+ 0 as T -+ for m > 0 (Bz(t,r) 1 in Section 3).

Thus assume A(t,-c) -+ 0 as T -+ 0 in which event, from (5.31) I there results (since B2/Bz = -az(.))

t az(4 (+:a1 (0~ (5.32) a,(.) Bz(o,~)C1 + + $(o)le do + 1 T

as T -+ 0, independently of t. But a,(a)/a,(o) -+ 0 as u -+ 0 and taking t small enough so that + $(a) 1 + @(o) (and B2(a,.r) 1 1) we have

1 (5.33) lim al(T)U(t,T) = T-+O rn

57 SINGULAR AND DEGENERATE CAUCHY PROBLEMS provided $(o) $ -1. This checks with Section 1.3 (for m > 0) 2m + 1 where 1/(1+$(0)) = or -U(t,T) -t 2m + Thus when T -2m . A(t,-c) * 0 as T * 0 with $(o) $ -1 we have by (5.30) as T * 0 YT(X,t,o) = -($(o)/(l+$(o))Z(X,t,o) (cf. (3.42) and the definition of Z in Lemma 5.8).

In general we cannot expect of course that A(t,T) -t 0 as

T * 0 or that $(o) $ -1 (cf. Section 1.3). However, as with 1 ?(y,t,.r), ye can show that YT(A,t,r) E L in T (with Y(A,t,.) absolutely continuous) and we follow again Carroll [8]. Thus m consider al(T)Y(X,t,-r) = 1 (-l)kal(~)Jk U(-,T) and define k=O for some upper limit T 5 2 .., t (5.34) Kn = 1 al(~)Jn = U(0,T)d-c 0 .., where Jn U(*,T) = (Jn U)(t,-c) (note here that (Jn U)(t,.c) =

0 by definition if T > t). We set again 1B2(u,~)1 5 c14 and lal(~)l5 klReal(T) so that

..,

(for Fubini-Tonelli theorems see McShane [l]). Similarly, setting [XIC 5 R and (c2 + Rc12) = c, we have (cf. (5.13) for J)

58 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

." 5 c,4k,cbF(t) 5 c14klcbF(b)

Upon iteration we obtain (cf. Section 3)

m m t and thus 1 (-1)"' = 1 (-l)n j al(~)(JnU)(i,T)dT converges n=O n=O 0 uniformly and absolutely. An elementary argument (cf. Carroll

[8]) then shows that al(*)Y(A,t,*)E L1 with

." t .., m (5.38) I al(T)Y(A,tyT)dT = 1 (-l)"Kn 0 n=O

." Lemna 5.14 If [all 5 klReal on [o,s] then T -+ YT(X,t,.r) = 1 -Z(A,t,T) + (a+@)(T)Y(A,t,T) E L for 0 5 T 5 t <- b and Y(X,t,*) T is absolutely continuous with Y(h,t,-r) = I Yg(X,t,<)d<. If 0 + = a'/,* E Co[o,s] and +(o) 4 -1 with h(t,T) = continuous with YT(A,t,o) = -(+(o)/( l++(o)))Z(A,t,o).

59 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Proof: The absolute continuity follows as in Lemma 3.9 and we recall that Y(A,t,o) = 0 by Lerlima 5.13. QED

Bounds for Y can be obtained as for Z. We simply multiply

(5.30), with T replaced by 4, by V(X,t,<), add this to its complex conjugate, and integrate in 4 from T to t, using the re- d lation - lY12 = 2Re(Y V), to obtain (recall that c1 + B = a1 +a2) d4 5

(5.39)

Recalling that c1 = 2 sup Rw2I we have, assuming Lemma 5.11,

t (5.40) IY12 5 (c YI2d4 + 1 1ZI2d4 T

5 c7b + (cl

Lemma 5.15 Under the hypotheses of Lemma 5.11 we have 2 - IYl (X,t,T) 5 c for 0 5 T r; t r; b and X > 0.

Proof: One applies Gronwall's lemma (Lemma 5.10) to (5.40). QED

Now let us go to the solution of the Cauchy problem (5.9) - (5.10), recalling that X can always be made larger than zero.

60 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Let Ls(E,F) be the space of continuous linear maps E -+ F with

the topology of simple convergence, i.e., Av -+ 0 in Ls(E,F) if

Ave * 0 in F for each e E E. We write L,(E) for LS(E,E) and 0 recall that OH = ff = \ ff(A)dv(A). Now Z(-,t,T) and Y(.,t,-r) belong to Ls(@D(A),ff), LS(ff), and, LS(eD(A)) since they are bounded by Lemmas 5.11 and 5.15 (on eD(A) we put the norm 2" 2 lle T II = j IleTll;(,)dv .I II~TllH(X)dvYwhere IleTIIqX) de- 0 c2E notes IleT(x)II If now eT eD(A) consider for example

A(eW)(X) = [Z(A,t,T) - Z(A,toyT)](~T)(X) = (AZ)(BT)(X). TO Show t -+ Z(*,t,-c)BT : [o,b] -+ 8D(A) is "strongly" continuous, and

hence that t -f Z(-,t,T) : [o,b] + Ls(eD(A)) is "simply" contin-

uous, one must show that as t + t -+ 0 IIA(OW)llff 0 and

IlhA(ew) Ilff + 0. But, for example, IIA(ew)llff + 0 means 2 1 IlA(e~)(X)ll~(~)-+ 0 in L (v) and since A(Bw)(X) + 0 v almost 2 2 everywhere (cf. Lemma 5.8) with IlA(Ow)(X) llff(X) 5 2c711eT11ff(h) E 1 L (v) (cf. (5.26) in Lemma 5.11) one can apply the Lebesgue dominated convergence theorem (cf. Bourbaki [3]) to conclude that IIA(ew)llH -+ 0 as t * to (note that lleTl12 is finite v H(X) almost everywhere). A similar argument applies to IlXA(ew) Ilff 2 1 since X lleT/12 E L (v); thus one has shown that t + Z(*,t,T) E H(X) CO(Ls(eD(A))) on [o,b]. Analogous reasoning, using (5.29) with Lemmas 5.8, 5.11, 5.13, and 5.15, leads to (cf. also Remark

5.22)

Lemma 5.16 Under the hypotheses of Lemna 5.11 we have, for

0 L T 2 t 5 b, t -+ Z(-,t,T) E CO(Ls(eD(A)) or Co(Ls(ff)) (and

61 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

CO(Ls(eD(A),H)) trivially); T -+ Y(*,t,-c) and T -+ Z(.,t,T) E

Co(Ls(ff)) or Co(LS(OD(A))); while t -+ Zt(*,t,T), t -+ a(t)Z,(*,t,T), and T + ZT(*,t,.r) E CO(Ls(eD(A),H)).

We wish to show now that in fact a/at(ew) = a/at(zeT) =

Z 8T in H. For fixed (A,T) one has evidently AZ/At - Zt(A,toT) = tt (1, [Zq(X,q,r) - Zq(X,to,~)]dq)/At. Hence in H(h) we have, v Y alm8st everywhere, in an obvious notation, Ah = (A(ew)/At) - t Zt(h,to,T)eT(h) = [( l/At)l AZndnleT(h) and consequently

Now IleTIIH(,l = I/eT(h)llH(hl is finite v almost everywhere and for (X,T) fixed we can make (by Lemma 5.8) lAZ,l 5 E for - 2 2 Irl-tol 5 6(~). Hence, for It-tol 5 6(~),IlAAl12 5 E ~~eT(A)~~H(h)

Y so that IIAhll + 0 v almost everywhere as t -+ to. Also by (5.27) " t we know that llAh112 5 [(2/At)(I (Cgx + ~~)~/~d~]~IleTlli(~)5 t ." 2 1 -+ (c17X + cia) IleT(h) IIH(h)* Therafore llAAIl2 0 in L (v) and 22 a/at(ew) = ZteT in H. Similarly one checks a /at (Ow), using estimates for a(t)Z,(h,t,T) and XZ(X,t,-c) indicated above (cf. also Lemma 5.9), to establish

Theorem 5.17 Under the hypotheses of Lemma 5.11, t -+ 2 Z(=,t,-r) E C (LS(eD(A)),H) and there exists a solution of the

Cauchy problem 5.9 (Problem 5.7) given by w(t) = e-'[Z(=,t,~)eT(*)] for T E D(A) (0 <- T 5 t 5 b).

62 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

One notes also that by (5.30), and Lemna 5.16 with refer-

ences thereto, T -+ (l/a(T))YT(*,t,T) cC0(LS(ff)) or CO(Ls(OD(A)) 1 1 for T small with T -+ Yf,-,t,~) E C (Ls(ff)) or C (L,(OD(A)) for

T > 0. Now we note that if F is barreled then any separately

continuous bilinear map E x F -+ G is hypocontinuous relative to bounded sets in E (see Bourbaki [2]). But a Hilbert space is barreled so the map u : (A,h) -+ Ah : Ls(F,G) x F -+ G is continuous on bounded sets Bc Ls(F,G). Since 0 5 T 5 5 S t 5 b is compact and the continuous image of a compact set is bounded

we may state for example that 5 -+ Y(*,t,t)ew(<), 5 -+ -+ Y(-,t,c)ew$). 5 -+ (46) + B(S))Y(-,~,S)@W~,5

-+ (5) are continuous (note Yg(*,t,S)0w5(S), and 5 Y(*,t,S)Bw 55 that Y5eW5 = (l/a(<))Y5 a 0w5 and ew(5) = ew(5,T) is given by

Theorem 5.17). The following formulas are then easily verified using Lemna 5.14 (cf. (3.18) - (3.20) and see Carroll [a] for de tai1 s ) .

(5.42) {Yewc5 + Y 0w Id5 = 0 55

where 0w satisfies (5.9). Hence multiplying (5.10) by Y we have

(cf. (5.30)). Hence 0w(t) is necessarily of the form Z(X,t,T)BT and we have proved

63 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Theorem 5.18 The solution of Problem 5.7 given by Theorem

5.17 is unique.

As before (cf. (3.22) and Theorem 3.14) we have (cf.

Carroll [8] for details and cf. also Remark 5.22).

Theorem 5.19 The unique solution of (L+Q(t)A)w = f E

Co(D(A)), W(T) = T E D(A), wt(-c) = 0 is given (under the hypo-

As in Definition 3.15 we will say that the problem 0 (L+Q(t)A)w = f, W(T) = T E D(A), w~(T)= 0, and f E C (D(A)) is uniformly well posed if w depends continuously in H on

(t,T,T,f). By hypocontinuity and Lemmas 5.8 and 5.13 we can state

Theorem 5.20 The problem (L+Q(t)A)w = f E Co(D(A)),

W(T) = T E D(A), and w,(T) = 0 is uniformly well posed under the hypotheses of Lemma 5.11 (0 5 T 5 t 5 b).

Remark 5.21 We can generalize some of the results of Sec- tion 1.3 to the case of L = a2/at2 + (a(t)+B(t)) a/at + y(t) in 2 2 2m+l place of 3 /at + (7)a/at; the details are in Carroll [8].

Thus one considers for 0 5 T 5 t 5 b e.g., Lw + Q(t)Ax * w = 0, I W(T) = T E S , and w~(T)= 0 with Condition 3.1 on A(y) ful- filled. It is in this problem that the analyticity of Z, Y, etc. in A is utilized (cf. Lemmas 5.8, 5.9, and 5.13).

64 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

2' One proves that the unique solution w E C (S ) of this general problem is given by w(t) = Z(t,-c) * T where Z(t,-r) = F-'Z(A(y),t,-r) and the problem is uniformly well posed.

Remark 5.22 In Carroll [lo; 111 some semilinear versions of the singular Cauchy problem were treated by spectral methods.

Let L be as above and construct, under the hypotheses of Lemma

5.11, resolvants Z(A,t,-r) and Y(A,t,-c) as above. Then consider in H

(5.45) (L+Q(t)A)w+ f(t,w) = 0; w(T.) = T E D(A); w,(T) = 0

(0 5 T 5 t 5 b). Following Theorem 5.19 this leads to the integral equation (where Z(t,T) = e-'Z(-,t,T)e and Y(t,T) = e-'Y(-,t,r)O belong to Ls(H)) (5.46) w(t) = Z(t,r)T - i: Y(t,S)f(S,w(S))dS = Tw Now parts of Lemma 5.16 can be somewhat improved by using estimates already established plus some stronger variations; for example one can show that (t,t) + Y(t,<) E Co(Ls(H,D(A 112 ))) and

1 112 t -f Z(t,.r) E C (Ls(D(A ),H) (cf. (5.27) for Z and for Y one 2 h can produce an estimate IYI (A,t,-c) 5 c/A for A > 0 and 0 5

T 5 t 5 b, stronger than that of Lema 5.15 - cf. also Chapter

3). One looks for fixed points of T using some modifications of a technique of FoiaT-Gussi-Poenaru [l;21. For example if we -, assume 5 +. f(S,v) = A112f(SyA-1v) E Co(H) for v E Co(H) with 1 [/;(S,V) -f(<,U)l[ 5 kl(<)wl(~~v-u~~)where kl E L and w1 is an

65 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

E Osgood function (i.e., dp/wl(p) = 00 for E > 0) while - 0 1 Ilf(C,V)ll -5 k2(E,)~2(jlvll)where k2 E L and w2 is a Wintner P function (i.e., dp/w2(p) = for E > 0) , then there JE exists a unique solution w E Co(D(A)) of (5.46) on [o,b]. Fur- thermore, this solution satisfies (5.45) with w E C2(H), w E

C1(D(A1l2)), and w E Co(D(A)) while (5.45) is uniformly well posed. The improved estimate on Y also leads to a stronger version of Theorem 5.19 where only f E Co(D(A’/2)) is required while 2 w E C (H), w E C’(D(A’/2)), and w E Co(D(A)). One can also add a nonlinear term to f(t,w) in (5.45) which generates a compact operator in H and obtain another existence result for (5.45).

1.6 EPD equations in general spaces. We go now to a technique of Hersh [l;21 in Banach spaces which was generalized by Carroll [18; 19; 24; 251 to more general locally convex spaces

(cf. a1 so Bragg [lo] , Carrol l-Dona1 dson [203 , Dona1dson 11 ; 51 , Donaldson-Hersh 161, Hersh [3]). Thus let E be a complete separated locally convex space and A a closed densely defined linear operator in E which generates a locally equicontinuous group T(*) in E (cf. remarks after Definition 6.3). This means that T(t) E L(E) for t ER, T(t) T(s) = T(t + s), T(o) = I, t +

T(t)e E Co(E) for e E E, and given any (continuous) seminorm p on E there exists a (continuous) seminorm q such that p(T(t)e)

5 q(e) for It\ 5 to< 00 (any to); also lim (T(t) - I)e/t = Ae as t + 0 for e E D(A) (dense). We refer here to Komura [ll and

Dembart [ll for locally equicontinuous semigroups or groups and

66 1. SINGULAR PARTIALOIFFERENTIALEQUATIONSOF EPD TYPE remark that it is absolutely necessary to consider such groups in

"large" spaces E in order to deal for example with growth properties of the solutions of certain differential equations (see also for example Babalola [l], Komatsu [l]a Lions [71 a Miyadera 111 ,

Oucii [ll, Schwartz [6] a Waelbroeck [l], Yosida [ll, etc. for other "general" semigroups - for strongly continuous semigroups in Banach spaces see Hille-Phillips [21). Let us consider now the problem

2m+l m = A2wm. (6.1) wTt + wt , w"(0) = eED(A2); wY(0) = 0 for wcC'(E). Following Hersh [l]one replaces A by a/ax and looks at

which is a resolvant equation as in Section 3 whose solution

^Rm = FRm is given by (3.6) with z = ty. We recall the Sonine integral formula (4.6) and pick p = -1/2, which corresponds to n 2p + 1 = 0 = n - 1, as the pivotal index (note p = 7 - 1 as in

Theorem 4.12). Clearly the unique solution of (6.2) with m =

-1/2 is R-'/2(*,t) = ux(t) = 1/2[&(x+t) + ~S(x-t)] so that, from (4.6), for m > -1/2 1 2r (m+l) (6.3) Rm(=,t) = (1-Am-7 vx(St)d6 r(1/2)r(m ; I'o 2m-- = cm] (1-6 ) 2 6(x-ct)d< -1 where c, = ,e(note that b(x?Ct) could be used in the

67 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

last expression in (6.3) because of symmetry involved in vx - cf.

(6.6)). Now one writes formally

(6.4) wm(t) = as the solution of (6.1) where for t fixed <, > denotes a pairing between the distribution Rm(*,t) E EI( of order zero and T(-)e 2 2 d E Co(E). Note here that for e E D(A ), T(-)e E C (E) with

T(x)e = AT(x)e = T(x)Ae, and of course t -+ Rm(*,t) E Cm(Ei) from

Section 1.3. If we write out (6.4) in terms of (6.3) there results

1 (6.5) wm(t) = C,,,/;~(~-S 2 ) m-- 2 < s(x-St), T(x)e > dS

which coincides with the formula established by Donaldson [l] using a different technique. We can also use the formula

2m--1 wrn(t) = 2cm / (1-S ) 2 cosh(A = cosh (ASt)e. To verify that (6.4) represents a solution of (6.1) one can of course work directly with the vector integrals (6.5) or (6.6) but we follow Hersh [l]and Carroll [18; 19; 24; 251 in treating (6.4) as a certain distribution pairing. Thus Rm(*,t),

R:(*,t), and R:,(-,t) = AxRm(*,t) are all of order less than or I equal to two in E with supports contained in a fixed compact

68 1. StNGULAR PARTtAL DIFFERENTIAL EQUATIONS OF EPD TYPE

A set K = Ix; 1x1 5 xol for 0 5 t 2 b < m and we let K = {x; 1x1 5

I xo + Bl for any f3 > 0. If S E E is of order less than o’r equal ZA1 to two with supp S C K we can think of S E C (K) (cf. Schwartz *2 ZA ill). Recall further that on K,C (E) = C g)E (see e.g., Treves E 2 111) where the E topology on C @ E is the topology of uniform 2’ I convergence on products of equicontinuous sets in (C ) x E . It is easy to show (see Carroll [18] for details)

a Lemma 6.1 The composition S +

tinuous for $ E C2(E) fixed. The map A @ 1 : C2(E) = COGE E

(defined by (A x I) 1 $i 8ei = 1 8’ei) is continuous. Let us indicate first the formal calculations necessary

to show that (6.4) is a solution of (6.1), while establishing at

the same time some recursion relations. Their validity is

essentially obvious upon suitable interpretation (cf. below).

Thus, from (6.4) and (3.11) transformed,

wF(t) = = T(ii3-T

-2(m+lJ-t 1 (1-5 2 ) m+- 2 T(Ct)A*edE

Similarly from (6.4) and (3.12) transformed

69 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

] = [wm-l (t) wm( t)l - , 2mt -

Then differentiating (6.7) and using (6.7) - (6.8) we obtain (6.1). The calculations under the bracket <, > can be justified using Lemna 6.1 and in order to transport A around in

(6.7) one can think of <, > as a distribution pairing over the

h interior of K (cf. Carroll [l8]). This proves, given A, T(t), 2 and E as above (e E D(A ))

Theorem 6.2 The equivalent formulas (6.4) - (6.6) give a solution to (6.1) for m 2 - 1/2 and the irecursion relations indicated by (6.7) - (6.8) are valid. The question of uniqueness for (6.1) had seemed to be rather more complicated than that of existence but a new theorem of Carroll [26] (Theorem 6.5) gives a satisfactory re- sponse. There are several types of theorems available (cf. Carroll [18; 24; 25; 261, Carroll-Donaldson [201, Donaldson 111,

Donaldson-Goldstein [2], Hersh [l]). The first one we discuss (for completeness) is based on a variation of the classical

"adjoint" method and simp1 ifies somewhat the presentation in

Carroll [18]; it is not as strong however, as Theorem 6.5.

Definition 6.3 The space E (as above) will be called A- I * adapted if E is complete and D(A ) is dense.

When E is A-adapted (A being the generator of a locally

70 1. SINGULAR PARTIAL DIFFERKNTIAL EQUATIONS OF EPD TYPE equicontinuous group in E) it follows (cf. Komura [ll)that A* * I generates a locally equicontinuous group T (t) in E . The requirement of completeness is a luxury which we permit ourselves for simplicity. Indeed, for the discussion of locally equicontinuous semigroups sequential completeness is sufficient (in view of the Riemann type integrals employed for example). Thus, in particular, if E is reflexive (not necessarily complete) then it is quasicomplete (cf. Schaeffer [l]),hence sequentially I * complete, as is the reflexive space E , and A will generate a * I * locally equicontinuous group T (t) in E with D(A ) dense (cf.

I Komura [ll).We remark also that if E is bornological then E is in fact complete (cf. Schaeffer [l]).We use completeness 2 2^ basically only in asserting that C (E) = C BE E but a version of this is probably true for E sequentially complete; the increase in detail and explanation throughout our discussion does not seem to justify the refinement however.

We consider now the resolvants Rm(-,t,.r) and Sm(*,t,T) of 2 22 Section 3 when A(y) = y with Ax = -A = -3 /ax (cf. also X Section 4). We recall (cf. Lemma 4.3) that Rm(-,t,r) and

Sm(*,t,T) belong to Ei, with suitable orders (exercise), and

I Lemna 3.13 holds with OM replaced by Exp OM = F E . Thus let wm(t) satisfy (6.1) with wm(0) = wF(0) = 0 and consider (for an

A-adapted E)

(6.9) um(t,<) =

71 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

"2 where el E D((A ) ) (dense); wm is an-y solution of (6.1), not necessarily arising from (6.4). Then from (3.17)

*I (6.10) uF(t,<) =

2m+l m *I = <(-Rm(-,t,<) + - S (-,t,<)),T (*)e> 5 2m+l m = -vrn(tL) + 7u (t,<) * where vm(t,<) = . Further, from (3.16), * (6.11) vm(t,<) = - = 5

- = -(A*)2 um(t,<)

Now take (E, El) brackets <, >E with u"(t,<) in (6.1), where t has been replaced by < and wm is an-y solution of (6.1 ) with zero initial data; then integrate in < from 0 to t (cf. (3.18) - (3.20)). We note first that t Ed< = -1 Ed5 (6.12) f: 0 2m t *2m (6.13) 1 EdE= I EdS= -Ed< 0

Consequently, using (6.10),

72 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Theorem 6.4 If E is A-adapted then the solution to (6.1), given by Theorem 6.2, is unique.

This uniqueness theorem has been included primarily to illustrate the adjoint method. If can be supplanted by the following recent result of Carroll [26] which was motivated by (and improves) a result in Carroll [18] based partially on a technique of Fattorini [l].

Theorem 6.5 For Rem > - 1/2 let wm be any solution of 6.1) with zero initial data and let A generate a locally equicont nuous group T(x) in E as above. Then wm(t) 5 0.

Proof: We note first that our existence-uniqueness calculations in Sections 3-4 are valid for suitable complex m

(Rem 2 - 1/2), as indicated partially in Section 5, but we will not spell out the details. Thus our demonstration is strictly justified only for real m 2 - 1/2. We use Rm(*,t, s) and 22 Sm(*, t, s) with Ax = - Ax = - a /ax as in the proof of Theorem 6.4. Define, with brackets as in (6.4),

(6.15) R(t,s) =

(6.16) S(t,s) =

73 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Then evidently (cf. (3.16) - (3.17) and the calculations leading to Theorem 6.4)

(6.17) RS(t,s) = -

2m+l m (6.18) Ss(t,s) = <-Rm(-,t,s) + S (-,t,s), T(*)w;(s)>

(6.19) (R+S),(t,s) = $,(t,s) = 0

Consequently $(t,o) = $(t,t) and recalling that S"(-,t,t) = 0 with m R (*,t,t) = 6 we have $(t,o) = 0 = $(t,t) = wm(t). (IED

We sketch here in passing some relations of (4.6) and (4.10) to the Riemann-Liouville (R-L) integral (cf. Carroll [18; 191,

Donaldson [l],and Rosenbloom [l]);questions of uniqueness for

(6.1) in E can be treated in this context but Theorem 6.5 includes everything known. The relation of EPD equations to R-L integrals was of course known by Diaz, Weinberger, Weinstein, etc. many years ago and some facts are of obvious general interest. We recall first (cf. Riesz [l])that the R-L integral of 1 f E Co or "suitable" f E L is defined for Re a > 0 by

t (6.20) (Iaf)(t) = m-1 io (t-s)a-'f(s)ds

74 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

One can obviously deal with f E Co(E) for example in the same manner and we will assume f is E valued in what follows. For

Re a > -p (p 2 1 an integer) the "continuation" of I' is given by

for f E Cp(E) or f E CP-'(E) with f(')(=) E L 1 (E) "suitably".

One knows that IaIB= IatBa except possibly when B is a negative integer B = -pa where IqPf= f(')a in which case f(o) = . . . = f('-')(o) = 0 is required in order that = in general, and Iois defined to be the identity. Evidently (d/dt)(I"f) = I"-1 f for any a. We record now some easily established facts.

Lemna 6.6 If f E Cn+'(E) with -n < Re a 5 -n + 1 and a nonintegral then

(6.22) (I"f)'(t) = (Ia-'f)(t) = -*-+ (Iafl)(t)

If f(o) = 0 or a = -n with f E Cn+'(E) then

(6.23) (Iaf)l = Iafl= Ia-lf

1 (when Re a > 0 (6.22) - (6.23) hold for f E C (E)). If f E Cn(E) with -n < Re a 5 -n + 1 or a = -n (or f E Co(E) when Re a > 0) then

(6.24) t(I"f)(t) = (I"(sf))(t) + a(Iat'f)(t)

Proof: Routine computation yields (6.22) - (6.23) and (6.24) follows by an induction argument. QED

75 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Now, referring to the Sonine integral formula (4.6) (cf. also (6.3) and (6.6)), we can write, after a change of variables, m- 111 f __ __ (6.25) Rm(*,-r1/') = I (T-S) 's 'R 2(-,s1/2)ds 0 where cm = .* as before. Let then

m m 1/2 (6.26) p(t) = with wm(t1/') = ; it follows from (6.25) that for Re m > -1/2 11 (6.27) Wm(t) = (I?W-')(t)

Setting im(t) = wm(t'/') it is easy to show (note that a/at'/* = zt'/' ( a/ at)

* Lemma 6.7 Given wm, w", and Wm as above, where wm satisfies the differential equation in (6.1), one has for Wm E C2(E)

(6.28) 4tqt + 4(m+l)wy = A2im

(6.29) 4tWyt - 4(m-l)W: = A2Wm

Remark 6.8 A previous uniqueness argument (cf. Carroll

[18; 191) for not necessarily A-adapted E proceeded from w" any solution-of (6.1) with zero initial data to Wm determined by 1 (6.26) and defined W-l" = I-m-'i W", together with the corresponding w-'". Then, using Lemma 6.6, given suitable (but excessive) hypotheses of differentiability and "regularity" on Wm,

76 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

W-l" was shown to satisfy (6.29) with index -1/2, and a unique-

ness theorem for the corresponding w-ll2 was established (weaker

than Theorem 6.5). This led to uniqueness for wm under suitable

hypotheses. One feature of the technique involved embedding Wm

in a "Weinstein complex'' determined by (cf. (4.10)

where p is an integer (0 5 p S a) and Re a > 0 or a = 0 (a and c1 are chosen so that -1/2 = m + a - R which means that if k + 1 3 3 7 < Re m < k + 7, k = -1, 0, 1, 2, . . ., then a = k + 7 - m with R = k + 2). (4.10) propagates initial values and (6.7) - (6.8) 1 hold. fl+a-p(o) and WT+a-P(o) (m+a-p 2 T) are critical in (6.30) and to study this- one notes from (6.28) - (6.29) that Wm corresponds to some w-~(for any index m) which we write in the form

(6.31 ) Wn ( t ) = i-n( t'l2)/r (n+l ) for some i-n(*) satisfying the differential equation in (6.1) with index -n. The connection is somewhat interesting and it is sufficient to indicate this for the crucial index 1/2 (note w1/2 = w-1/2 and in (6.26) W-'"(O) = 0 does not automatically t insure W-'/2(o) = 0). Thus differentiating (6.31) with index 1/2 and noting the obvious difference between ft(t'/') and f(t1/2)t =

(d/dt)f(t'/') we obtain (cf. (6.26))

(6.32) Ft1 -1/2w1/2(t1/2) + t 1/2w1/2(t1/2) = ;-l/2(t1/2)t

77 SINGULAR AND DEGENERATE CAUCHY PROBLEMS whereas from (6.8) with index 1/2 we have (since a/at 112 =

(6.33) 2t 1/2( p),

Consequently from (6.32) - (6.33) we have

(6.34) w-1’2(t1’2) = 2t1/2Y2( w t1/2)

and thus given W-’/~(O) = 0 there results

Therefore i-1’2(t1’2)t-+ 0 as t -t 0 since w;’/’(<) -+ 0 which

means by (6.31) that W;/‘(t) = W-l/‘(t) -t 0 as t -f 0, as desired.

In any event we have existence and uniqueness determined by Theorems 6.2 and 6.5 and some new growth and convexity theorems will follow, with realistic examples, of a type first developed by Carroll [18; 19; 251. Thus, referring to (6.6) - (6.7), and taking now m real so that cm > 0 for m > -1/2 we have under the hypotheses of Theorem 6.2

Theorem 6.9 Let E be a space of functions or equivalence 2 classes of functions and assume e E D(A2) with cosh (At)A e >_ 0 (m >_ -1/2). Then wm(*) is a nondecreasing function of t.

Proof: From (6.6) - (6.7) we have for m t -1/2

m tA’ (6.36) wt(t) = 2(m+17 w“+’(t) 1 1 m-+ 2 cosh(A6t)A e dS 2 0 m+l QED

78 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONSOF EPD TYPE

22 Next we recall that (from Section 4) Li = a /at t

2m + a/at = (2m) t -2(2m+1) a2/a2(t-2m) for m 0 while Lot = t t-2 a2/a2(10g t). The differential equation in (6.1) can be

written as LkP = A2wm so we have from (6.6) with m > -1/2 real

1 ni-- 1 (6.37) Lmwtm (t) = 2cm 1 (1-5 cosh(A5t)A2e d5 0

Theorem 6.10 Under the hypotheses of Theorem 6.9, ~"(0)is a convex function of t-2mfor m + 0 and of log t for m = 0.

Proof: The result for m = -1/2 is obvious since w-'l2(t) =

cosh(At)e and the rest follows from (6.37). QED

Example 6.11 Let E = Co(IR) with the topology of uniform

convergence on compact sets. The operator A = d/dx generates a

locally equicontinuous group (T(t)f)(x) = f(x+t) which is not equicontinuous. If f L 0 then evidently T(t)f 2 0 for t E R so to fulfill the hypotheses of Theorems 6.9 - 6.10 we need only II find e = f such that A2f = f 2 0; such functions are abundant,

but of course they increase as t + and we do indeed want to work in "large" enough spaces E to permit such growth.

Example 6.12 Let E = C"( R3) x C"( R3) where C"( R3) has

the Schwartz topology and recall the mean value operators p,(t)

and Ax(t) defined in Section 2 which we extend as even functions

for t negative. Define

01 A0 (6.38) A = (A );A2= (0 A 1

79 SINGULAR AND DEGENERATE CAUCHY PROBLEMS where A is the three-dimensional Laplacian. Then A generates a locally equicontinuous group T(t) in E acting by convolution

(i.e., T(t)? = T(t) * 1) determined by

One notes again that T(t) is not equicontinuous. From (6.39) it follows that

(6.40) cash At = u,(t) + gAt2 2 Ax(t) so that cosh At 7 2 0 provided ? 2 0 and A2? 2 0. Given 7 = fl (f ) this means that fi 2 0 with Afi 2 0. Thus in order to apply 2 Theorems 6.9 - 6.10 we want ? such that A 2' f 2 0 and A4? 2 0 (i.e., 2 Afi 2 0 and A fi 2 0) which is possible e.g., when fi(x) = i exp lyjxj. Again the necessity of using "large" spaces E is ap- parent. We note also from (6.39) that T(t) and T(-t) behave quite differently relative to the preservation of positivity. Analogues of these growth and convexity theorems for other "canonical" singular Cauchy problems appear in Carroll [18; 19; 251 (cf. also Chapter 2).

1.7 Transmutation. The idea of transmutation of operators goes back to Delsarte and Lions (cf. Delsarte [l], Delsarte- Lions [2; 31, Lions [l; 2; 3; 4; 51) with subsequent contributions

80 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE by Carroll-Donaldson [ZO], Hersh 131, Thyssen [l;23, etc. The subject has many yet unexplored ramifications and is connected with the question of related differential equations (cf. Bragg-

Dettman, loc. cit., Carroll-Donaldsoc [20], Hersh [3]); we expect to examine this more extensively in Carroll [24]. This section therefore wi 11 be partia1 ly heuri stic and the word "sui tab1e" will be used occasionally when convenient with precise domains of definition unspecified at times.

One forumlation goes as follows. Let Dx = a/ax and Dt = a/at and consider polynomial differential operators P = P(D) and

Q = Q(D) (D = D, or Dt). One says that an operator B transmutes

P into Q if (formally) QB = BP. B will usually be an integral operator with a function or distribution kernel and in fact one often assumes this a priori, although it is perhaps too restric- tive (cf. Remark 7.3). One picks a space of functions f (the choice is very important) and considers the problem

k (7.2) @(x,o) = f(x); Dt@(x,o) = 0 x,t 2 0 where 1 5 k 5 m - 1 with m = order Q. In order to define B we put @(o,t) = (Bf)(t) and writing $(x,t) = P(Dx)@(x,t) there re- sul ts

81 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

One may wish to extend f and P(D)f appropriately for x < 0 when desirable or necessary. There results formally

(7.5) uJ(o,t)

- and evidently it have constant coefficients. It is however necessary to suppose that (7.1) - (7.2) have a unique solution in order to well define B; the same uniqueness criterion applies to (7.3) - (7.4) but this will often be trivial if f is chosen in the right space.

Theorem 7.1 Suppose the problem (7.1) - (7.2) (resp. (7.3) - (7.4)) has a unique solution + (resp. $) with f in a suitable space so that (7.5) makes sense and define generically (Bf)(t) =

+(o,t). Then B transmutes P(D) into Q(D). 2 Let us indicate some simple applications. Let P(D) = D and Q(D) = D so that (7.1) - (7.2) become

We prolong f as an even function and take partial Fourier trans- A A 2^ A A forms x + s (Ff = f) to obtain +t = -s @ with i(s,o) = f(s). A A 2 The solution is +(s,t) = f(s) exp (-s t) and if

82 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

(7.7)

2 it is known that FxR(x,t) = exp (-s t) (cf. Titchmarsh [Z]).

Hence (R(*,t) * f(-))(x) = @(x,t) and recalling that f is even there results (7.8) @(oyt)= (Bf)(t) = c f(S)K(S,t)dS (cf. Carroll-Donaldson [ZO]). An easy calculation shows that

I I I1 (7.5) will hold provided f (0) = 0 and of course f, f , and f must have suitable growth at infinity.

Now there is no difficulty in extending the transmutation

idea to vector functions f with values in a complete separated

locally convex space E. For example suppose that A is a suitable

(closed, densely defined) operator in E with g(A) a reasonable operator function and let P(D)w = g(A)w where w takes values in

E (actually in D(g(A)) since A may not be everywhere defined).

If B transmutes P into Q then formally Q(D)Bw = BP(D)w = Bg(A)w = g(A)Bw provided that w satisfies the conditions necessary for the

transmutation and that Bw E D(g(A)). In this event u = Bw will satisfy Q(D)u = g(A)u and there arises the general question of when this situation can prevail. If B is an integral operator of

Riemann type with a function kernel then g(A) can be passed under the integral sign and this part of the question is trivial up to the point where initial or boundary values arise. As an example let us consider the case (cf. Carroll-Donaldson [ZO])

83 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

2 2 (7.9) P(D)w = D w = A w; w(o) = uo; wt(o) = 0

2 (7.10) Q(D)u = Du = A u; u(0) = uo

If A2 = B these are of course wave and heat equations and the connecting formula is well known, but we will derive it via transmutation. First formally we use (7.8) transmuting D2 into D to obtain

(7.11) u = Bw = w(<)K(S,t)dS c I and we note that wt(o) = 0, corresponding to f (0) = 0, as required in (7.8). Suitable growth of w, wt, and wtt will be as- 2 2 sumed. Further A Bw = aA w for t > 0 while for t = 0 one must require u E D(A~). Consequently 0

Theorem 7.2 Given w a solution of (7.9) it follows that u 2 given by (7.U) satisfies (7.10), provided uo E D(A ) while w, wt, and wtt grow suitably at infinity.

Such results are not new of course, at least in more c assi- cal forms, and we refer to Bragg-Dettman, loc. cit., Carrol -

Donaldson [ZO], Hersh [3], Lions, loc. cit., and references there.

Remark 7.3 One can relax (7.2) and study not well posed

Cauchy problems (or other problems) which lead to a unique solution @ (cf. Carroll-Donaldson [ZO]); for example growth conditions on @ and f could be imposed (see here Carroll [24]). Lions in his extensive investigations, chooses spaces of functions f

84 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE where the transmutation is an isomorphism in the sense that there is an inverse transmutation B-' sending Q(D) into P(D) (i.e.

PB- = B-lQ) but this seems to be an unnecessary luxury in genera , although obviously of great interest (cf. Section 4.3). The choice of spaces for f is in any event of paramount importance for the transmutation method to work. If B is an integral operator with a function or distribution kernel one can begin with this as a stipulation and then discover the differential problem which the kernel must satisfy (not necessarily a simple Cauchy problem), and we refer to Carroll [24; 251, Carroll-Donaldson [20], Hersh [3], Lions, loc. cit., etc. for further information (cf. also (7.25)). We will conclude this section with a version of Lions trans- 2 2 = = + mutation method sending P(D) D into Lm D t D (cf. also Section 4.3). Thus we look for a function $(x,t) satisfying

+ (7.12) ?m 1

@xx = 4tt + tt;

I and again we extend f to be even with f (0) = 0. This is of course the one dimensional EPD equation whose solution is given by (6.3) as (cf. Theorem 4.2)

t 1 (7.14) $(o,t) = (Bf)(t) = 2cmt-2m 1 (t -T ) m-Ff(-r)d.r 0

85 SINGULAR AND DEGENERATE CAUCHY PROBEMS which agrees with Lions [l; 2; 3; 51. Consequently

t 1 (7.15) u = Bw = 2c t-2m (t -T ) m-Pw(-c)d-c m 1 0 is the vector version of (7.14) and we have

Theorem 7.4 If w satisfies (7.9) with uo E D(A2) then u given by (7.15) satisfies Lmu = A 2 u with u(o) = uo and ut(o) = 0.

I Proof: Evidently (Bw)(o) = w(o) and (Bw) (0) = 0 (cf. Lions [l]where an explicit verification of (7.5) is also given - the I condition w (0) corresponding to f (0) = 0 is crucial here as in t (7.8) and (7.11)). Again A2 can be passed under the integral sign in (7.15) as before in (7.11). QED

Remark 7.5 Changing variables in (7.15) we obtain (6.6) when W(T) is written as W(T) = cosh Awe. Actually in this case there is also an inverse transmutation operator B-l but we will not go into details here (see Lions, loc. cit.. for an exhaustive study of the matter). In Lions [3] 2 it is also shown how to transmute P(D) = D into Q(D) = Mm = 2 D D + p(t)D q(t) and vice versa, and we will sketch + t + some of the calculations here (p and q are assumed to be con-

tinuous) (cf. also Hersh [S]). Thus let $(x,t) be the solution of

(7.16)

(7.17) $(x,o) = f(x); $t(~,~)= 0 (x,t 2 0)

86 1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

I w'here f is extended as an even function with f (0) = 0. Then

again the transmutation operator B is determined by (Bf)(t) =

r$(o,t) and satisfies formally the transmutation relation MmB =

BD2, while to construct B Lions uses the following ingenious

"trick." He sets

(7.18) B(cos xs)(t) = 0(s,t)

and from the transmutation relation plus the fact that (Bf)(o) =

f(o) we obtain

2 (7.19) MmO(s,t) + s 0(s,t) = 0 with e(s,o) = 1, while et(s,o) = 0 automatically (.for unique

solutions e(s,t) see Section 5). Assuming now (we omit verifi-

cation) that B is defined by a (distribution) kernel b(t,x) with

support in the region 1x1 5 t, which is legitimate here but is in part incidental to the transmutation concept, one obtains,

since f(x) = cos xs is even, and b(t,*) will be even (verifica-

tion follows from (7.25) for example)

t (7.20) 0(s,t) = 2 b(t,x) cos xs dx = 2 b(t,x) cos xs dx 0

from which follows, by Fourier inversion in a distribution sense,

(7.21) b(t,x) = 1 O(s,t) cos xs ds 0 2 Theorem 7.6 The transmutation of D into Mm is determined

by an operator B with kernel defined by (7.21), where €Iis known.

87 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Further properties are developed by Lions [3; 51 and a general transmutation theory will be developed in Carroll [24].

Hersh's technique (cf. Hersh 131) is based on special functions and kernels and leads formally to the same result as that of

Theorem 7.6. Let us sketch this for completeness. Thus let w(t,A) be the solution of

2 (7.22) Mmw + A w = 0; w(o,A) = 1; wt(oyA) = 0 while u satisfies (cf. (7.10))

2 2 (7.23) D u + A u = 0; u(0,A) = 1; ut(o,A) = 0

THen u(t,A) = cos At and Hersh proposes a formula

m (7.24) w(t,A) = b(t,x)u(x,A)dx = b(t,x) cos Ax dx ,m

Since (7.22) holds we have, provided b(t,-) has compact support with reasonable smoothness ,

(7.25) Mmb(t,x) = bxx(t,x)

This shows symmetry of b(t,x) in x and (7.24) becomes (cf. (7.20))

(7.26) w(t,A) = 2 [ b(t,x) cos Ax dx

Again (7.25) can be solved, as can (7.19), by the methods of

Section 1.5. Defining now B by (Bf)(t) = b(t,x) f(x) dx we 2 L have formally from (7.25) MmB = BD . The idea then is to replace 2 A by suitable operator functions -g(A).

88 Chapter 2

Canonical Sequences of Singular Cauchy Problems

2.1 The rank one situation. In this chapter we will develop the group theoretic version of the EPD equations studied in Chapter 1. This leads to many new classes of equations and results parallel to those of Chapter 1 as well as to some interesting new situations. Moreover it exhibits the main results of Chapter 1 in their natural group theoretic context. The ideas of spherical symmetry, radial mean val'ues and Laplacians, etc. inherent in EPD theory have natural counterparts in terms of geodesic coordinates and one can obtain recursion relations, Sonine formulas, etc. group theoretically. The results are based on Carroll [21; 221 in the semisimple case and were anti- cipated in part by earlier work of Carroll [18; 191, Carroll-

Silver [l5; 16; 17) and Silver [l] for some semisimple and Euclidean cases. The group theory is "routine" at the present time and relies heavily on Helgason's work (cf. Helgason [l; 2; 3; 4; 5; 6; 7; 8; 9; 101) but one must of course refer to basic material of Harish-Chandra (cf. Warner [l; 21 for a sumnary) as well as lecture notes by Varadarajan, Ranga Rao, etc.); other specific references to Bargmann [l], Bargmann-Wigner [2], Bhanu-Munti [l], Carroll [27; 281, Coifman-Weiss [l], Ehrenpreis-

Mautner [l], Flensted-Jensen [1], Furstenberg [l], Gangolli [l], Gelbart [l], Gelfand et al, [l; 2; 3; 41, Godement [l], Hermann

89 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

[l], Jacquet-Langlands [l], Jehle-Parke [l], Kamber-Tondeur [l], Karpelevic" [l], Knapp-Stein [l], Kostant [l], Kunze-Stein [l; 23, Lyubarskij [l], Maurin [l], McKerrell [l], Miller [l; 23, Naimark [l], Pukansky [l], RGhl [l], Sally [l; 23, Sims [l], Smoke [l], Stein 113, Takahashi [l], Talman [l], Tinkham [l], Vilenkin [l], Wallach [l], Wigner [l], etc., as well as the fundamental work of E. Cartan and H. Weyl, are to be taken for granted, even if not mentioned explicitly. The basic Lie theory is developed somewhat concisely in this section; for a rather more leisurely treatment we refer to Bourbaki [5], Carroll [23], Hausner-Schwartz [l], Helgason [l; 2; 31, Hochschild [l], V Jacobson [1], Loos [l], Serre [l; 21, Tondeur [l], Zelobenko Ell, etc. We will start out with the full machinery for the rank one semisimple case, following Carroll [21; 221, and later will give extremely detailed examples for special cases. This avoids some repetition and presents a "clean" theory immediately; the reader unfamiliar with Lie theory might look at the examples first where many details and definitions are covered. We delib- erately omit the treatment of invariant differential operators acting in sheaves or in sections of vector bundles even though this is one of the more important subjects in modern work (some references are mentioned above). The preliminary material will be expository and specific theorems will not be proved here. The Eucl idean group cases have been covered in Carroll -Si 1 ver [15; 16; 171 and especially Silver [l] so that we will only give

90 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS a few remarks later about this at the end of the chapter; the basic results are in any event included in Chapter 1. Thus let G be a real connected noncompact semisimple Lie group with finite center and K a maximal compact subgroup so that V = G/K

-.-d is a symmetric space of noncompact type. Let g = k + p be a

Cartan decomposition, a c p a maximal abelian subspace, and we will suppose until further notice- that dim a = rank- V = -1. One sets A = exp a, K = exp k, and N = exp n where n =

IsA for A > 0 where the gA are the standard root spaces corresponding to positive roots a and possibly 2a in the rank one case. 1 One sets p = - cm A for X > 0 where mA = dim gA and we pick an 2x 1 element Ho E a such that a(Ho) = 1. Thus p = (H ma + m2a)a and we can identify a Weyl chamber as a connected component a+C I a c a with (o,m) in writing a(tHo) = t where u E R corresponds * to u E a by u(tHo) = Ut. The Iwasawa decomposition of G is G =

KAN which we write in the form g = k(g) exp H(g) n(g) where the I notation at = exp tHo is used. Let M (resp. M ) denote the centralizer (resp. normalizer) of A in K so that the Weyl group I is W = M /M and the maximal boundary of V is B = K/M (thus M =

I {k E K; AdkH = H for H E a) and M = {k E K; Adka ca) - see the examples for specific details). There are natural polar coordinates in a dense submanifold of V arising from the decomposition G = G+K (A+ = exp a+) provided by the diffeomorphism

(kM,a) -+ kaK : B x A+-+ V (one could also work with the decomposition G = KAK). Thus the polar coordinates of ~(g)=

91 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

ak2) E V are (klM,a) where IT : G +. V is the natural map.

given v = gK E V and b = kM E B one writes A(v,b) = -1 2 k) and the Fourier transform of f E L (V) is defined by

Y Ff = f where (using Warner's notation)

Y r (1.1) f(u,b) = 1, f(v) exp (iu+p)A(v,b)dv * for u E a and b E B. All measures are suitably normalized -in this treatment. This sets up an isometric isomorphism f - f 2 * between L (V) and the space L2(a: x B) (cf. here below for a+) with inversion formula

Here the 1/2 comes from the order of the Weyl group (which is two) and c(u) is the standard Harish-Chandra function. For a general expression of C(M) we can write (cf. also (3.14)) c(u) = I(ip)/I(p) where (cf. Gindikin-Karpelevic [l],Helgason

[3; 81, Warner [l;21)

Here B(x,y) = I'(x)r(y)/I'(x+y) is the Beta function and one defines (v,a) in this context in terms of the Killing form * B(-,*) as B(H ,Ha) where for X E aC, HA E aC is determined by V * A(h) = B(j,H) for H E a (note here that aC is the space of R linear maps of a into while aC is the complexification of a which is formally the set of all sums A + iu for A,u E a). Then

92 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS * a+ is the preimage of a+ under the map A +. HA and is a Weyl * chamber in a . Specific examples of c(p) will be written down * * later. Now a /W - a+ (.recall W = (1,s) where sH = -H for H E a or equivalently sat = a-t = at’) and one can write

2 where, for $ E L (B),

The quasiregular representation L of G on L2 (V), defined by

L(g)f(v) = f(g-’v) decomposes in the form r (1.5) L= Ja*,W ‘plC(w) dp where L acts in H by the same rule as L. L is in fact ir- P Fc 1-1 reducible and unitary and is equivalent to the so-called class one principal series representation induced from the parabolic

subgroup MAN by means of the character man -f a” = exp ip log a.

We recall here also the definition of the mean value of a function @ over the orbit of gr(h) = gu under the isotropy h subgroup Iv= gKg-’ at v = .rr(g). Thus writing M C$ = Mu$ we have

(recalling that r(h) = u)

93 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(cf. Helgason [l])when D E D(G/K). The symbol D(G/K) denotes the left invariant differential operators on G/K which are defined as follows. If $ : G -+ G is a diffeomorphism and f is a function on G one sets f$(g) = (f $-')(g) while if D is a -1 linear differential operator we write DQ : f + (Of$ )$. Thus

G or on V = G/K. Writing akg = gk for g G and k E K, the T space of differential operators on G such that D = D and D"k =

D is denoted by D(G/K); this is identical with the space of linear differential operators on G/K (called left invariant) T such that D = D (the condition Dak = D is automatic on G/K - proof obvious). Now the zonal spherical functions on G are * defined by the formula (u E a ) ......

.., and we can write by invariant integration 41 (9) = $ (gK) (ac- .., u u .., tually the $ (9) are K biinvariant in the sense that $ (9) = .., *- u u $u (kgk)) and K is unimdular - cf. Helgason [l],Maurin [l], Nachbin [2], Wallach [l],or Weil [l]- and thus f(ikk)dk = I, .., f(k-')dk = 1 f(k)dk. It is known further that $,(g) = ..,K K C$I C$I (g-l) (cf. Harish-Chandra [2] or Warner [l; 21) while the -u E C"(G/K), and are characterized by their eigenvalues AD = @u X ($ ) for D E D(G/K), with $u(n(e)) = 1, plus the biinvariance uu.., of (I~.We now demonstrate a lemma which has some interesting consequences.

94 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

h Lemma 1.1 The Fourier transform of 1.1 = MU E E’(V) (u = - A- .rr(h)) is given by FMh = (h) and if h = kak with a E A+ then 1-I (M‘f)(v) = (Maf)(v) so that (Muf)(v) depends only on the radial

component a in the polar decomposition (kM,a) of u = .rr(h).

h Proof: The action of M or MU as a distribution in E’(V) is h h determined by (cf. (1.6)) = (14 $)(w) where w = r(e) (e

being the identity in G). Thus in (1.6) take Q = exp (ill + p)

A A A(w,b) where b = kM. Since A(ekhK,kM) = -H((kh)-’;) one has by

the unimodularity of K (cf. (1.8) and comments thereafter)

(Mh$)(w) = exp [-(ip+p)H(h -1 k -1,- k)dk (1 -9) I K

= j exp [-(ip+p)H(h -1 kk)dk = j e-(i’+p)H(h-lk)dk K K .., .., - h = $ (h-’) = 0 (h) = FM -1-1 1-I h One notes here that M = MU works on functions in V such as h exp (i’+p)A(v,b) = $ and is precisely (1.1) with f re- h placed by 14‘ = M and $ = exp (iu+p)A(w,b); no integration over

V is involved since v = w in $, only a distribution evaluation

A- is of concern here. Finally we note that with h = kak as above

there follows from (1.6)

(1.10) (Mhf)(v) = J f(gkiaK)dk = f(gkaK)dk = (Maf)(v) QED K i, This leads to a symmetric space version of theorem of

Zalcman [l],generalizing an old formula of Pizzetti; here G/K

95 SINGULAR AND DEGENERATE CAUCHY PROBLEMS is of arbitrary rank and we refer to Helgason [l ; 23 or Warner [l; 23 for general information on higher rank situations.

Theorem 1.2 (Pizzetti-Zalcman). Suppose the operators

Ak with eigenvalues Xk generate D(G/K) as an algebra; then local ly

(1.11) (MUf)(V) = bqu, Ak)f)(V)

00 where $,(u) = 1 + 1 Pn(u,X (4 )) is expressed in terms of its n=l kv eigenvalues as $ (u,Xk). P

Zalcman [l], in an Rn context, has generalized this kind of theorem considerably .

h Proof: One notes from Helgason - [l] that (M @,)(v) = $X(gk.rr(h))dk = ;Jgkh)dk = zX(g)$X(h)and the local expres- K i, O3 sion (Muf)(v) = ([1 + 1 Pn(u,Ak)]f)(v) holds. Consequently n=l - applying this local expression to we obtain $X(h) = m 1 + 1 P,(U,A~($~))= $,(u) and (1.11) follows. One should n=l remark also that the polynomials P n are without cons tan t terms.

2.2 Resolvants. The objects of interest n a generalized

EPD theory are the radial components of a basis for the H v spaces of (1.4), multiplied by a suitable weight funct on, the result of which we will denote by @(t,u) (cf. Carroll [21; 221, Carroll- Silver [15; 16; 171, Silver [l]) and we mention that m can denote

96 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS a multiindex here. First we remark that V is endowed with the

Riemannian structure induced by the Killing form B(*,*) (but the Riemannian structure does not play an important role here, e.g., $(*,*) could serve - see Example 3.2) and for rank V = 1, D(G/K) isgenerated by a single Laplacian A, determined by - the standard Casimir operator C in the enveloping algebra of g

(cf. Remark 3.15 about C). We look at the radial component AR of

A, passing this from the coord nate t in at E A to r(A) in an a A - obvious manner, and setting Mt = M with Ro(t,p) = FMt = $,,(at) one obtains an eigenvalue equation (cf. Helgason [5])

[of. + (ma+mZa)cotht Dt + mZatanht D 141 (2.1) tu

The solution of (2.1), analytic at t = 0, is e.g.,

A 2 (2.2) ^Ro(tyu) = (Iu (exp tHo) = F(G,B,y, - sh t)

"0 "0 where evidently R (0,~)= 1 and Rt(o,p) = 0 (6 = (ma+2m2a+ 2ip)/4,

B = (m +2mZa-2iu)/4, and y =(ma+m2a+1)/2). The general EPD a "0 situation involves embedding R in a "canonical" sequence of "m llresolvants" R (t,p), for m > 0 a positive integer or a multiindex, such that the resolvant initial conditions

"m "m (2.3) R (0,~)= 1; Rt(oYp) = 0 are satisfied. There will also be "canonical" recursion relations

97 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

“m between the R of indices differing by k1 or k2 which will arise group theoretically from considering a full set of basis elements in the H spaces. 1-I 2 Thus we must first determine a basis for L (B) and this is well known (thanks are due here to R. Ranga Rao for some helpful information). We let {IT~,V~Iwith dim VT = dT be a complete set of inequivalent irreducible unitary representations of K and let M VT CVT be the set of elements fixed by M. One knows by a result of Kostant [l]that dim Vy = 1 or 0 in the rank one case

(cf. also Helgason [5]) and for the set T E T where dim V! = 1 we let wi be a basis vector for VTM with wY(1 2 i 5 dT) an orthonormal basis for VT under a scalar product < >T. Then for example the collection of functions (T E T)kM -+ T is 2 known to be a basis for L (B) and we define (cf. (1.4) with v = atK and b = kM)

n- (2.4) $;,,(atK)

= El 9T(BT:-ip:a -1 )w T tj where BT E HomM(VT9$) is determined by the rule B‘w: = 61,s (Kronecker symbol) so that BTrT(k)-1 w.T = T = J JT <~~(k)-’w;,w;>~ (note that J $(k)dk = I (I $(km)dm)db and K BM 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

H(g-’km) = H(g-’k) with nT(km) = nT(k)nT(m)). The above are Eisenstein integrals as defined in Wallach [l]and the

A. (a K) are the radial components of basis elements in H M,T t FC (cf. below). It is possible to obtain an explicit evaluation of these functions, using results of Helgason [5; 111 as follows.

One defines (cf. Helgason 151)

for x = gK. Then it is proved in Helgason [5; 111 (to whom we gratefully acknowledge some conversation .on the matter) that

where Y (x) is defined by ,T (2.7) yA,T(x) = j e (-iA-p)H(g-’k) < wl,T aT(k)wi >T dk K h A Thus recalling the polar decomposition (kM,a) + kaK we have

Theorem 2.1 General basis elements in H me FC

-1^-1 (2.8) (iatK) = e (iu-p)H(at k k) .,T dk ;’MrT 1 K

A AT = d-ll2T fT,J-’.(katK) = ~Y-FC,T(atK)

In particular one has G’ (a K) = (a K) = Y (atK) VaT t ;’ 1-I.T t -’,T since ti1 = (this “collapse” was observed in ,j J T 1T special cases by Silver [l]). A M Now let KO = {nT,VTl for T ET (i.e., dim VT = 1) and following Helgason [5] we use a parametrization due to Johnson and

99 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Wallach (see Helgason [5] for references and cf. also Kostant

h [l]).If mZcl = 0, KO = {(p,q)> with p E Z+ and q = 0; if mZa = .. 1, KO = {(p,q)> with (p,q) E Z+ x Z where p f q E 22,; if mZa =

h 3 or 7, KO = {(p,q)> with (p,q) E Z+ x Z+ where p & q E 22,. One sets R = (ip+p)(HO) and then (cf. Helgason [5; 111)

Theorem 2.2 The radial components of basis elements in H IJ are given by

R+p-q+l -m2a ma+m +1 = c '1.I STthPt ch-'tF(-,--p R+ + + ,th2t)

where th = tanh, ch = cosh, and F is the standard hypergeometric function.

Note here that our R is the negative of the R in Helgason

[5; 111 where R = (iA-p)(H ) = (-ip-p)(H ) (cf. here our notation 0 0 in (1.1)) and recall that a(Ho) = 1 with at = exp tHo. If one now sets dZa = -4q(q+mZcl-1) and da = -p(p+m +mZa-l) + q(q+mZcl-l) it is shown in Helgason [5] that Y(t) = Y (atK) also satisfies -1-I ,T the differential equation

100 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

(2.11) ytt + (ma+in2a)coth t Yt + m2a tanh tit

+ [d sh-2t + d2ash-22t + p(Ho) 2 + v(Ho) 2 19' = 0 a where sh = sinh, and we have used the identity coth 2t = 1 hcoth t + tanh t). Recalling that p ."(-in2a +m 2a ) one sees that the trivial representation 1 E T, corresponding to p = q = 0, gives rise to (2.1), so that (since c = 1) we should have .w R+I-M, m +m +I (atK) = +(at) = ch-'t F(L/2, 2a , a 2a , th t). This y-lJ ,1 is borne out by the Kummer relation F(a,b,c,z) = (l-z)-aF(a,c-b, 2 c,z/z-1) with z = -sh t, a = 6 = R/2, b,= 8, and c = y, so that

(l-~)-~= ch-'t and c - b = y - B = (ma+2+2iv)/4 = (R+1-m2a)/2;

." hence Y (atK) = +(a ) as indicated. -v,1 t

2.3 Examples with = 0. Now to construct resolvants h A Rm(t,v) = RPsq(t,p) from the Y (a K) one multiplies the -!J,T t by a suitable factor in order to obtain the resolvant ini- y-v ,T ." tial conditions (2.3) and to produce I$I (a,) when p = q = 0. v These requirements are not alone sufficient to produce the "canonical" resolvants since one needs to incorporate certain group theoretic recursion relations into the theory which serve to

"split" the second order singular differential equation for the

A Rp,q arising from (2.11) into a composition of two first order equations (cf. here Infeld-Hull [l]). Even then we remark that the resolvants will not be unique since one can always multiply

^Rm(t,p) by a function $,,, E C 2 such that $,(o) = 1, $i(o) = 0,

This will simply give a different second order I+ 0 (t) E 1.

101 SINGULAR AND DEGENERATE CAUCHY PROBLEMS singular differential equation for the new resolvant and different splitting recursion relations but the resolvant- initial conditions will remain valid and the reduction to 0 (a ) for m = Ut 0 is unaltered. Thus we will choose the simplest form of resolvant fulfilling the stipulations imposed while referring to these resolvants and equations as canonical.

Example 3.1 Take the case where ma = 1 , mZa = 0, da = 2 -p , d2, = 0, and p = 1/2. This corresponds to G = SL(2,IR) with

K = SO(2) and the resolvants are given in Carroll [18; 19; 223,

Carroll-Silver [15; 16; 171, Silver [l]as

where 5 = cht and Pim denotes the standard associated Legendre function of the first kind. Now we recall a formula (cf. Snow

111, P. 18)

2 2 2 Setting z = th t in (3.2) and noting that 1 - th t = sech t we have fi = sech t and 1 - n/(l+Z) = 5-1/<+1 with 5 = cht as above. Now one observes that (t,U) in (3.1) is symmetric

102 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

i.n !J (and t) and this is exhibited for example in the form of the resolvant (3.1) when one uses the formula (cf. Robin [l]) 1 1 2-mshmt m+-+ip m+--ip 2 2 (3.3) P-m (cht) = F( , '2 ,m+l,-sh t) ip- z Hence (3.1) can be written (recall that F(a,b,c,z) = F(b,a,c,z)) 1 1 "m m+T+ip m+q-ip 2 (3.4) R (t,u) =F( , , m + 1, -sh t)

Now, returning to (3.2) and (2.9) - (2.10), we write for

T - (p,~)with ct = id2, B = 1/2(p + 1/2), and R = iu + 1/2

R 1 t sech t)- 2(ct+B) = c thPt ch- t( -!J,T

1 1 5- 1 = c -!J ,T shPt(y)-'-PF(ip + -2 + p, ill + 7, p+l, 5+1)

Rewriting the first equation in (3.1) with p replaced by -!J in the right hand side we obtain

(3.6) im(t,p) = ($!-) -R-m F(z+1 ip,z+1 ip+m,m+l,-) 5-1 5+1

Then identifying p with m we can say from (3.5) - (3.6)

h (3.7) RP(t,p) = sh-Pt Y (a K) -!J,T t

103 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

For completeness we check the differential equation satisfied by

the Rp of (3.7), given that (2.11) holds. An elementary calcula-

tion yields then

which agrees with previous calculations (cf. Carroll [21; 223,

Carroll-Silver [15; 16; 171, Silver [l]).The canonical recursion

A relations associated to the Rp of (3.1), (3.4), (3.6), or (3.7)

can of course be read off from known formulas for the associated

Legendre functions for example but they can also be obtained group theoretically (cf. Example 3.5 - Theorem 3.8) by using a full set of basis elements in the H spaces. For now we simply list them P in the form

i! + 2p coth t ^Rp = 2p csch t iP-'

Evidently the composition of these two relations yields (3.8) and

this is the sense in which we speak of "splitting" the resolvant

equation (3.8).

Example 3.2 We give now explicit matrix details to clarify

Example 3.1. (The Euclidian case can also be treated group

theoretically as in Carroll-Silver [15; 16; 171 and especially

Silver [1] and a simple example is worked out at the end of this

chapter; the results of course agree with those of Chapter 1.)

104 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS - Thus - let G = SL(2,lR) and K = SO(2) with Lie algebras g = sR(2,R) and k = so(2). We recall that G is connected and semisimple, consisting of real 2 x 2 matrices of determinant one,- while the matrices in the compact subgroup K are-- orthogonal; g consists of real 2 x 2 matrices of trace zero and kcg is composed of skew symmetric matrices. We write V = G/K and set

1 O 11., Y=z1 (3.10) X = 7 10 - so -that k = lR Z and we write p = {JRX +.lRY) for the subspace-- of g spanned by X and Y. --One has- a Cartan --decomposition g = k + p (of vector spaces) with [k,k] c k, [p,p] c- k, [k,p] cpand the Cartan involution 8 -: 5 + n -t 6 - Q (6 E k, n E p) is a Lie algebra automorphism of g. Recall here that if P = exp p then G =

PK is the standard polar decomposition of g E G into a product of a positive definite and an orthogonal matrix. The Killing form B(5,n) = trace ad5 adn (=4 trace CQ) is negative definite " - on k and positive definite on p (with B(k,p) = 0). One checks easily that X and Y form an orthonormal basis in p for the 1 scalar product ((6,~)) = p(5,n);we repeat that the 1/2 factor is of no particular significance here in constructing resolvants, etc. and is used mainly to be consistent with some previous work and with the exposition in Helgason [l].

Now setting Xa = X + Z, X-a = X - Z, and Ha = 2Y we have a standard (or "canonical") triple (cf. Serre [l])

105 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

where the root subspaces g, = RX, and g = RX are character- -a .-- -, ized by the rule gA = (5 E g; ad H< = X(H)S for all H E a = RY), while the map a : tY -+ t determines then an element (called a * * root) in the dual a (note that ,(Ha) = 2 and R N a = Ra).

Here a is a maximal abelian subspace of p and in this case a = - .-- .-. a is also a Cartan subalgebra of g. Set now n = g, = .-- - {I g,,, > 01, N = exp n, and A = exp a. The Iwasawa decomposition of G can be written (0 2 e < IT)

(3.13) g = k a n = exp BZ exp t Y exp exa ett

which we express as before in the form g = k(g) exp H(g)n(g). Next we set M = {+ [b :]I for the centralizer of A in K and 1 I4 = M u {+ will be the normalizer of A in K (thus M = [-: I -.-- {k E K; AdkH = H for H E a and M = {k E K; Adk a c a) ). Again I W = M /I4 has order two and B = K/M is essentially K with the angle variation cut in half. We recall that for v = gK E V and b = kM E B A(v,b) = -H(g-lk) and here p = 1/2 lmAA (A>O) equals

106 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

1 "* p. Thus p(tY) = t/2 and for 1-1 E R, 1-1 E a is determined again -* by the rule u(tY) = tu(Y) = 1-1t (such l~ of course exhaust a and 2 ct - 1-1 = 1). The Fourier transform Ff of f E L (V) is given by (1.1) and a+ is defined as in Section 1, as is the scalar prod- -* uct (X,v) = B(HA,HV). When 1-1 E a is characterized as above then 1 * since B(Y,Y) = 2 it follows that H = pYand obviously a+ can 1-1 "*" thus be identified with (0,~). Evidently for 1-1, v E a one has

Example 3.3 A geometrical realization of the present situation can be described in terms of G acting on the upper half ab plane Im z > 0 by the rule g z = (az+b)/(cz+d) for g = (c d) E

G. Then K is the isotropy subgroup of G at the point z = i

(i.e., K i = i) and the upper half plane can be identified with G/K under the map gK -+ g i (cf. Helgason [l]). In geodesic polar coordinates (T,e) at i the Riemannian structure is 22 22 described by ds =d-c + sinh .cde where we use the metric tensor determined by gv(<,q) = $(C,q) for 5, q E p. If w = r(e) (which corresponds to i under our identification since r(e) = K) then, denoting left translation in G by T the Riemannian structure is g' determined by gV((T )*<,(T~)*T-I) = gw(S,q) (for the notation (T )* 9 9 see e.g. Kobayashi-Nomizu [l; 21). The geodesics on V through w =.rr(e) are of the form y : s -+ (exp sC)w and if 5 = aX + BY E p the square T' of the geodesic distance from w to (exp <)w = 2 .rr(exp 5) is equal to a2 + B (since X and Y form an orthonormal basis for p under the scalar product ((<,q)) = 1/2 B(5,n)) and

107 SINGULAR AND DEGENERATE CAUCHY PROBLEMS the polar angle is determined by tan 8 = B/a. Many calculations can easily be made using standard formulas and one can describe the action g i for example in general (relevant material for the computations appears in Gangolli [l], Helgason [l; 2; 31, and Kobayashi-Nomizu [l; 21). We will spare the reader the results of our computations since they are essentially routine. Let us however remark that the volume element dv in (1.1) can be written in this example as dv = shT d.r d8 (up to a normalization factor).

Example 3.4 First we refer back to the formula for c(v) given after (1.2) and mention that in general (cf. Helgason [2; 31, Warner [Z])

Y where n = On = g-ahere, K = exp n, and dii is normalized by c(-ip) = 1. Here 8 is the Cartan involution 8 : 5 + rl + E - rl

Y for 5 E k and rl E p and one observes that it is an involutive

Y g -+ automorphism-.-. of such that (c,rl) -B(€,,Orl) is strictly positive definite on g x g. In terms of Beta functions we obtain

Since (cl,a) = 1/2, (ip,a) = ip, and (p,a) = 1/4 there results (cf. Bhanu Murti [l])

108 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS - Thus we can recover f(v) from f(u,b) using (3.16) and (1.2).

Example 3.5 We now want to show how the resolvants (3.1) can be obtained by a cumbersome and "classical" method in order to further confirm (.3.7) as a legitimate resolvant (the equivalence of (3.1) and (3.7) has already been established). The technique has a certain interest and moreover we will obtain a full basis for H while showing how the recursion relations (3.9) 1-1 can be obtained group theoretically. Some of the more tedious calculations will be omitted. We remark that Theorem 2.1 does not seem convenient here to obtain a full set of basis elements in H u since the unitary irreducible representations of K = SO(2) are one dimensional of the form ein4 (see Vilenkin [l]).Thus

hhh h first we can remark that atk0 = k a n where a = as with

(3.17) es = (ch t t sh t cos 0)

(cf..Helgason [l]and Warner [l;23). Thus pH(atke) = p(sY) = s/2 and for ~(sV)= us one has (iu-p)H(atkg) (i1-1- I I 2)s ill- (3.18) e =e = (ch t -+ sh t COS 0)

Now, given basis vectors $m(b) = exp im(0-n) for example in LL(B) (recall that B - K with the angle variation, in our notation of (3.13), cut from (0, 4n) to (0, IT)), we get basis vectors in H by means of (1.4) (the factor of exp (-im) is inserted for u convenience in calculation). In particular we obtain "radial objects" when v = atK in (1.4) and since A(gK,kOM) = -H(g-'kO) we

109 SINGULAR AND DEGENERATE CAUCHY PROBLEMS consider H(ailkg) = s'Y in (1.4). where s' is determined as in

(3.17) but with t replaced by -t. Thus the radial part of the basis vactors in H will be u

I e (iu-p)s Yeimede (3.19) 9 = u

I -imr TT iv- = 7 e irned0 21T 1 (ch t - sh t cos 0) -lT

iv- 1 - (ch t) 'm .o '

(see Vilenkin [l]). In order now to produce resolvants R%l (tau) as in Section 2.2 from Pm - 1/2(ch t) of (3.19) one must first a0 consider the growth of these functions as t +. 0 and introduce suitable weight factors in order to satisfy the resolvant initial conditions (2.3) (note that a/at =(~~-1)l/~3/22). Let us recall in this direction that (cf. Vilenkin [l]and Robin [l])

r (ktmtl ) r ('R-mtl ) R (3.20) P:am(~) = r2(R+i) pm,o(z)

where P! is the standard associated Legendre function of the first kind. Then we can exhibit the resolvants in the form

110 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

z+l R-m z-1 F( -R ,m-R ,m+l ,-) = (T) z+l where z = ch t, R = -1/2 + ip, rn 2 0 is an integer, and F denotes a standard hypergeometric function (cf. Carroll-Silver [15; 16;

171 and Carroll [18; 191). Recalling that r(R+m+l)Pim(z) = r(a-m+l)P~(z) for m 2 0 an integer, we remark in passing that, for m E C not a negative integer, resolvants can be determined by the formula

(which reduces to (3.21) when m is an integer) and this should be * taken as the generic expression for Rm in this case (cf. (3.1)). Evidently the ^Rm of (3.21) satisfy (2.3). We recall the next well known formulas

d P! mz (3.23) (~~-1)"~-+ ,/2 P! = (a+m) (R-m+l )P!-' dz (22-1)

(cf. Vilenkin [l]and Robin [l])and these, with (3.22), lead to 12 (3.9), where p is replaced by m (note that R(R+1) = -(B+ u )); consequently (3.8) will follow as before.

The most interesting fact however about (3.9) is that these recursion relations have a group theoretic significance (cf.

111 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Carroll-Silver [15; 16; 171) and the variable t in (3.8) - (3.9) may be identified with the geodesic distance T of Example 3.3.

For purposes of calculation it is convenient to consider

(3.25) X+ = -X + iY; X - = -X - iY; H = -2iZ and to note that (H, X+, X-) form a "canonical" triple relative -. - Y to the root space decomposition gt = kt + tX+ + CX - , kt = (CH, in the sense that

(3.26) [H,X+] = 2X+; [H,X-] = -2X ; [X+,X-] = H

(one notes the difference here between (3.12) complexified as -. 9Q; = Q;Ha + t$ + tX, and the decomposition above relative to (H,X+,X-) based on (3.26) - such decompositions occur relative to any Cartan subalgebra such as (CH or ICY as indicated in Serre

[l]for example). It will be useful to exploit the isomorphism

Q between SL(2,R) and the group SU(1,l) of unimodular quasiuni- A 1i tary matrices given by Q(g) = g = m-lg m where m = (i l). Thus ab g = (y E) + 6 = (6:) with a = 1/2[a+6+i(B-y)] and b = A 1/2[~+y+i(a-6)] while det g = det g = 1 (cf. Vilenkin [l]).

There is a natural parametrization of SU(1,l) in terms of gener-

h alized Euler angles (I$,T,$) so that any g E SU(1,l) can be

written in the form

112 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS where 0 I cp < ZIT, 0 5 T < m, and -2n I: $ < ZIT; this will be ex-

h pressed as g = (cp,~,$). In particular, setting wl(s) = exp sX, wz(s) = exp sY, and w3(s) = exp sZ, we have Qwl(s) = (o,s,o),

Qw,(s) = (IT/Z, and (s) =(s,o,o) = (o,o,s). We note s, -v/Z), Qw3 that to,.,$)-’ = (IT-$,T,-IT-@) and (~~,o,o)(o,T,o)(o,o,$) = A A Ah (+,T,$) while if g1 = (0, T~,O) and g2 = (cp2,~2,~) then gigz =

(cp,T,$) with

sh -rz sin cpz (3.28) tan cp = ch T~ sh -r2 cos cp + sh T 1 ch 2 Tz ’

ch T = chTl ch T~ + sh T~ sh T~ cos cpZ ;

sh T, sin cpz tan ’ = sh -r1 ch T~ cos cpZ + ch T~ sh ‘c2

A Now going to the notation of Example 3.3 with p = exp (ax + BY),

T‘ = a2 + $, and tan e = @/aan easy calculation shows that A Q(p) = (O,-r,-O). Consequently the geodesic polar coordinates of

A h A ~(p)= pK can be read off directly from the Euler angles of Q(p).

Now the representation L of G on H (cf. (1.5)) induces !J u a representation, which we again call L of ion dense subspaces u’ W E H of differentiable functions by the rule uu

Recall here that L,,(h)f(n(g)) = f(h-lv(g)) = f(v(h-lg)) and to apply (3.29) for 5 = X, Y, or Z one wants then to differentiate f(n(gi(s))) with respect to s where gi(s) = wi(-s)g = wil(s)g

113 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(and v = T(g)). We work in the geodesic polar coordinates

A A (~(s),e(s)) of .rr(g(s)) = T(p(s)k(s)) = ~(p(s))and consider therefore Q(g(s))= Q(;(s))Q(k(s))= (e(s), ~(s),-e(s)) (o,o,ds)) = (W,~(s), +(s)-e(s)) = (e(s),T(s) ,$ts)). In particular Q(gl (s)) = Q(exp(-sX))Q(g)=(o,-s,o) (8,T,o)(o,o,u-8) where Q(g) = Q(p)Q(k) = (O,T,-e)(o,o,u). Then one uses (3.28) to compute (o,-s,o)(O,T,o) with $ = e(s), T = ~(s),T~ = -s,

T~ = T, and 0, = e (evidently $(s) is of no interest here).

S imi 1arly Q( g2 (s ) ) = (~/2,-s ,-~/2) (8 ,T ,-8) ( o ,o, U) = (~/2,o ,o) (o,-s,o)(~-T/~,T,o)(o,o,u-~)and (3.28) can be applied to the middle two terms. Finally Q(g3(s))= (-s,o,o)(B,T,o)(o,o,u-8) which lends itself immediately to calculation. We write H+ = 1 Lv(X+), H- = L (X ), and H3 = 4 (H) and using the relation 2v u- I d/ds f(T(s),e(s))ls=O = fT T (0) + fe 0 (0) a routine calculation yields (cf. Carroll-Silver [16])

Proposition 3.6 In geodesic polar coordinates (.r,8) on V one has

(3.30) H+ = e-ie [-i coth T a/ae + a/aT]

(3.31) H- = eie [i coth T a/ae + a1a-c-j

(3.32) H3 = i a/ ae

From genera known facts about irreducib e unitary principal series representations of G, or SU(1,1), and their complexifica- tions (see e.g. Miller [l; 21 or Vilenkin [l]) - other

114 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS representations contribute nothing new in this context - it is natural now to look for "canonical" dense differentiable basis vectors in Hu of the form fk(T,e) satisfying

(3.33) H+f: = (m-R)f:+

H3fi = mfi where R = -1/2 + ip(u > 0) and m = 0, kl, k2, . . . . (It seems excessive here to go into the general representation theory of SL(2, R ), or SU(1,l); the references cited here -and previously at the beginning of the chapter - are adequate and accessible.) It is then easy to verify, using (3.30) - (3.32), that the following proposition holds.

Proposition 3.7 Canonical basis vectors in Hu can be taken in the form

R (3.34) fi(T,e) = (-l)m exp(-im8) P (ch T) 0 ,m where R = -1/2 t ip(u > 0) and m = 0, 21, k2, . . . .

In view of (3.20) and the definition (3.21) (or (3.22)) of

8 we can write for m 2 0, z = ch T, and R = -1/2 + iu

Using (3.30) - (3.31) one can then easily prove (cf. Carroll- Silver [16])

115 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Theorem 3.8 The "canonical" relations (3.33) for H, and H - "m are equivalent to the recursion relations (3.9) for R . In particular the variable t in Example 3.1 can be identified with the geodesic distance T, from w = K to k a K. UJt

Proof: The recursion relations (3.9) (and hence (3.8)) follow immediately from (3.30) - (3.31) and (3.34) - (3.35). This suggests of course the identification of T and t but one can give

A a computational proof also. We write k,ak2 = pk so that kla = 3 n n pk4 and for p = exp (aX+BY) we obtain from Example 3.3 the dis- 2 n tance r2 = a2 + B between w = T(e) = K and pK together with angle measurements cos e = a/T and sin 8 = B/T. Set now kl =

a = aty and k4 = kg, using the notation of (3.13); there kJ, result a number of equations connecting the variables

(Tyt,$y$yayf3)ythe solution of which involves the identification of t with T. Thus the radial variable t in the natural polar expression (k My a ) for u = T(h) = kUJatK is in fact the geo- J,t A desic radius T of T(h) = pK. The actual calculations are somewhat tedious but completely routine so we will omit them. QED

We note now that the second equation in (3.9) can be written

in the form (shZmt @)' = 2mshzm-lt. 8-l which yields, in view of (2.3),

sh2m-1 In-1 (3.36) shZmt im(tyu)= 2m 1: rl R (rlyil)drl One rewrites the integrand in (3.36), in terms of (3.36)m-1 and

116 2. CANONICAL SEQUENCES OF SINGUALR CAUCHY PROBLEMS

integrates out the rl variable; iterating this procedure we obtain

Theorem 3.9 If in 2 k 2 1 are integers the "Sonine" formula

(3.37) shZmt km(t,p) = r m-k+l r k jt(cht-chrl)k-'

is a direct consequence of (3.9). In particular for m = k we have

t (3.38) shZmt @(t,p) = 2% I (cht-chn)m-lsho io(q,u)do 0

Remark 3.10 Referring back to the formula for the zonal spherical functions (1.8) and using the notation of (3.17) -

(3.18) we see that (cf. (3.19)) 1 ,., 27T ip- (3.39) +v(at) = & I (cht + sht cose) 'dB = Po ,(cht) 0 1v-7 1 ip- - where P l(cht) = Po '(cht) is the standard Legendre function. 'p-7 SO .., We recall also by Lemma 1.1 that FMat = FMt = + (a ), which Ut equals io(t,p). -l*m Using (3.38) we can define Rm(t,-) = F R (t,v) E EI(V) and integrating in E'(V) (cf. Carroll 1141) we have for m 2 1 an integer,

(3.40) shZmt Rm(t,-) = Zmm (cht-cho)m-lshn M drl rl

We recall that in the present situation D(G/K) is generated by the Laplace-Beltrami operator A which arises from the Casimir

117 SINGULAR AND DEGENERATE CAUCHY PROBLEMS ... operator C in the enveloping algebra E(g) (here A corresponds to

2C due to our choice of Riemiannian structure determined by 222 1/2B(-,*) and C = 1/2(X +Y -Z ) - see Remark 3.15 below on Casi-

-3 mir operators). In E(g B) we have then also 2C = ($1)' + $X+X- + X-X,) and in geodesic polar coordinates (cf. Proposi- tion 3.6)

(3.41) A = H: + $H+H- + H-H,) = a 2 + coth T a/a-c

222 + csch T a /a0

We note here that an alternate expression for A is given by A = .. H+H + H' H3 (cf. Carroll-Silver [15; 16; 171 and Silver -3- Ill).

Theorem 3.11 For v fixed (Muf)(v) depends only on the geodesic radius T = t of u = n(h) = k a K and can be identified 9t with the geodesic mean value M(v,t,f).

Proof: This can be proved by tedious calculation based on the coordinates introduced in previous examples but it is simply a consequence of Lemma 1.1 and Theorem 3.8. Indeed, setting

A AA Ah g = k ak with h = k a k we have v = n(g) = k aK and u = n(h) = 12 1 t2 1 klatK with a (3.42) (Muf)(v) = (M tf)(v) =(Mtf)(v)

AA AA We note that the distance from v = klaK to k aka K is the same as 1t

118 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

A the distance from w = n(e) = K to katK = (exp E) w = pw, which as ndicated in Theorem 3.8, will simply be T = t. Hence as k ~n var es the points klak.rr(at) describe a geodesic "circle" around nh k aw of radius T = t; (3.42) is, upon suitable normal zation V' 1 already provided, the definition of M(v,t,f). QE D

Remark 3.12 The validity of the Darboux equation ( .7) for geodesic mean values M(v,t,f) and D = A in spaces of constant negative curvature, harmonic spaces, etc. has been discussed, using purely geometrical arguments by Ghther [l],Olevskij [l], and Willmore [l](cf. also Fusaro [l],G'u'nther [2; 31, Weinstein

[12], and Remark 5.6).

We can now define, for m 2 1 an integer, a composition (or convolution) of Rm(t,.) E E'(V) with a function f(-) on V by means of (3.40). Thus for v = n(g) we write

(3.43) where < , > denotes a distribution pairing as in Lemma 1.1

(cf. He gason [2] for a similar notation). Then, setting

(Rm(t,= # f(*))(v) = um(t,v), we have from (3.40) a Sonine formula

(3.44) sh2mt urn(t,v)

I Since F = R(R+l)FT wi h R = -1/2 + iu when T E E (V) (cf.

Ehrenpre s-Mautner [1]) there follows from (3.8), (3.9), (3.43)

119 SINGULAR AND DEGENERATE CAUCHY PROBLEMS and the definition of u" = u"(t,v) = um(t,v,f)

2 Theorem 3.13 For m 2 1 an integer and f E C (V) the function um(t,v,f) = um(t,v) = (Rm(t,-) # f(*))(v) satisfies

(3.45) u; = qq)-sh [A-m(m+l)]~"+~

(3.46) uy + 2m coth t um = 2m csch t urn-'

m (.3.47) utt + (2m+l) coth t uy + m(m+l)um = Aum

(3.48) um(o,v) = f(v); uT(o,v) = 0 where (3.45), (3.47), and (3.48) hold for m 2 0.

Proof: The notation here directs that (AMt # f)(v) =

Av(Mtf)(v) = (MtAf)(v) (see e.g. (1.7) and recall also that A =

6d + d6 in the notation of de Rham [1] is formally symmetric) while (3.48) is immediate from the definitions and (3.45) for example. One can also prove Theorem 3.13 directly from the definition (3.44) and the Darboux equation (1.7) without using the

Fourier transformation (cf. Carroll-Silver [15; 16; 173). Indeed

(3.46) follows immediately from (3.44) by differentiation while

(3.45) and (3.47) result then by induction using (1.7) and the 22 radial Laplacian DU = A = a /at + coth t a/at (cf. (3.41)) for m = 0 together with (3.46) (see Carroll-Silver [16] for details).

QED

Definition 3.14 The sequence of singular Cauchy problems

120 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

(3.47) - (3.48) for m 2 0 an integer will be called a canonical sequence. The canonical resolvant sequence corresponding to

(3.8) under inverse Fourier transformation is (cf. also (2.3))

(3.49) RTt + (21~1)coth t RY + rn(m+l)Rm = ARm

m (3.50) Rm(o,*) = 6 ; Rt(0,*) = 0 where 6 is a Dirac measure at w = .rr(e). Similarly from the equations (3.9) we have

(3.51) Rt = 2(m+l)sh [A-m(m+l)]Rm+’

(3.52) RY + 2m coth t R” = 2m csch t Rm-’

Remark 3.15 We recall at this point a few facts about .-. Casimir operators for a semisimple Lie algebra g (cf. Warner [l]). .-. If X1, X is any basis for g let gij = B(Xi,X.) where . . ., n E J B(=,*) denotes the Killing form and let gij be the elements of the matrix (g’j) inverse to (g..). The Casimir operator is then 1J defined as

This operator C is independent of the basis {Xi} and lies in the .-. center Z of the enveloping algebra E(gt). Another formulation .“- can be based on a Cartan decomposition g = k + p and a e stable - .-. .-..-.-..-. Cartan subalgebra j of g (i.e., j = j f) k + jfl p = j- + jp); ... k such j always exist. Let a basis Hi (1 <, i 5 m) for j-. and a k

121 SINGULAR AND DEGENERATE CAUCHY PROBLEMS basis H.(m+l

When G has finite center with K a maximal compact subgroup then the operator in D(G/K) determined by C is the Laplace-Beltrami operator for the Riemannian structure induced by the Killing form

B(*,-) (see Helgason [l]and cf. (3.41)).

Example 3.16 We consider now the case where G = SOO(3,1) =

SH(4) is the connected component of the identity in the Lorentz group L = SO(3,l) and K = SO(3) x SO(1) Z SO(3). The resulting

Lobazevskij space V = G/K has dimension 3 and rank 1. In R4 G consists of so called proper Lorentz transformations which do not 2 2 reverse the time direction. Thus set [x,x] = -r (x,x) = -xo + 222 + x2 + x3 and think of xo as time. Then G corresponds to 4 x 4 x1 matrices of determinant one leaving [x,x] invariant and preserving the sign of xo. In particular if n4 denotes the interior of the 2 positive light cone r (x,x) > 0, xo > 0, then G : R4 + R4. The

LobaEevskij space V can be identified with the points of the 2 "pseudosphere" r (x,x) = 1, xo > 0 (which lies within R4). If coordinates

122 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

x1 = r sh sin €I2 sin x3 = r sh €I3 cos €I (3.55) e 3 e 1’- 2;

x2 = r sh €I3 sin €I2 cos €I1; x0 = r ch €I3 with 0 5 €I1 < ZIT, 0 5 O2 < IT, 0 5 e3 < m, and 0 5 r < m are prescribed in R4 then coordinates ei(i = 1,2,3) can be used on V and 2 the invariant measure dv on V is given by dv = sh e3 sin O2lIdei

(i= 1,2,3). We will now follow Takahashi [l]in notation because of his more “canonical” formulation but refer also to Vilenkin

[l]for many interesting geometrical insights. .., .., Thus as a basis of the Lie algebra g of G we take

0’ 01 Io O 1000 00 0

(3.56) Y1 = Y Y2 = Y 0000 00 0

000 0 00 0)

000 1’ 00 0’

0000 010 Y3 = 3 3 ‘12 = 0000 -1 0 0

1000, 00 0,

000 0’ 0. 0 0’ 000 1 00 0 ‘1 3 Y ‘23 000 0 00 1

0 -1 0 0, 0 -1 0, SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Note that if E denotes the matlrix with 1 in the (p,q) position Pq (pth row and qth column) with zeros elsewhere then Y - + P - EoP E (p = 1,2,3) and X - - E (p

Y.., Y have a Cartan decomposition g = k + p and K = exp k. Take now a = RY1 as a maximal abelian subspace of p and set X = Yp + X P 1P (p = 2,3) so that

’0 0 1 0’

0010 0001 (3.57) x2 = ; x= 1-1 0 0 3 0000

,o 0 0 0, 1 -1 0 0, - Then writing n = RX, + RX an Iwasawa decomposition of g is -- .., L 3 - given byg = k+atn and writing A = exp a with N = exp n one has G = KAN. We will use the notation at = exp tY1 so that

‘ch - sh t 0 0

sh t ch t 0 0 (3.58) at = 0 0 1 0

(0 0 0 1

One can take Y1 = Ho in the general rank one picture with a(Ho) =

1 since [Yl,X2] = X2 and [Y, ,X3] = X3. Thus a, which is the 1 only positive root, has multiplicity two and p = -in a = a with 2a p(tY1) = t. We recall that M = (kek; AdkH = H for H E a1 and that Mi = (kEk; Adka C a). Since exp AdkH = k exp Hk-’ we see

124 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS that AdkH = H for H E a implies k exp Hk-l = exp H while AdkH E a for H E a implies k exp Hk” = exp Hi for HI E a. It is immediate that M = {exp consists of matrices

0 A (3.59) m = 0 1 ; m E SO(2)

I whereas M = M U {exp vXl2, exp ITX~~),where exp ITX (p = 2,3) I 1P sends at to a,t. Thus W = M /M has the properties indicated before in the general rank one situation and B = K/M can be identi- 23 fied with the two sphere S c R . This identification can be spelled out in at least two ways and we will adopt the one leading to the simplest integral expressions later. Thus, following

Takahashi [l], we write u = exp 0X12 and v = exp I$X23 for 0 5 0 I$ 0 5 IT and -IT < I$ 5 IT (in particular v E M). Then any k E K can 0 be written in the form k = m u v for m E M so that if m = v 00 JI then kll = cos 0, k12 = sin 8 cosb k13 = sin 0 sin I$, kZ1 =

-sin 8 COSI), and k31 = sin 8 sin+ One can pass the map k + x = 2 (kll,k21,k31) : k + R3 to quotients to obtain a map K/M + S where k + v u M and x + (JI,0); here -IT < JI < TI and 0 < 0 < IT $0 2 on the domain of uniqueness. The invariant measure on S corresponding to this identification is given by ds = (41~)-lsined0dJI and the functions

125 SINGULAR AND DEGENERATE CAUCHY PROBLEMS with c = [(2n+l)(n-1n)!/(n+m)!]"~, -n 5 m 5 n, and n = 0,1,2, m ,n 22 . . ., form a complete orthanormal system in L (S ) (cf. Talman [l]and Vilenkin [l]).

We can now give explicit formulas for a canonical set of radial objects in H, (cf. (1.4)). Thus let v = atK and b = kM with k = v+ue and consider (cf. (3.19))

A IT IT (iP-l)H(a;'k) (e,$) sin 8 dOd$ (3.61) @;"(atK) = -IT I0 e 'n ,m

It follows from general formulas in Takahashi [l]that exp H(a-'v u ) = ch t sh t cos 8 and therefore (3.61) becomes t $0 -

(3.62) "m0,' n (atK) = @h~ 'n (atK) = Zn(t) 1-I ,

IT = 2'0 n I (ch t - sh t cos e)ip-lPn(cos e) sin e de 2o IT since 1 eimd$ = 0 for m f 0 and P:(cos 0) = Pn(cos 0). We note -IT ." "0 that (3.62) reduces to R (t,w) = FMt = @ (a,) when n = 0 since 1-I c = 1 and Po(cos 0) = 1 whereas, setting k = v, u v and re- 030 - +@@ @ is an even function of t, the expression for calling that u io(t,p) can be written in the form (cf. Takahashi [l])

(iu-l)H(ai'k) "0 (3.63) R (t,p) = I e dk K

126 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

= 1 r(cht-sht cose)iv-l sinede = Zo(t) JO v

- 2ipsh t [(cht + sht)” - (cht - ~ht)~’]

= sinpt/vsht

To evaluate (3.62) in general one has recourse to various formulas for special functions and we are grateful here to R. G. Lange- bartel for indicating a general formula for such integrals in terms of Meijer G functions (cf. Luke [l]). The details are somewhat complicated so we simply state the result here. Thus, since

We note that for n = 0 this formula yields (cf. Magnus- Oberhettinger-Soni [l])

11 (3.65) Z;(t) =qsh-’t P-’ l(ch t) i v-7L = (L)sh-’t[(ch t + sh t)iv- (ch t + sh t)-iv] 21 v

= (-) 21U1 sh-’t(e ivt - e-ivt) in accordance with (3.63). Further let us remark that (cf.

Robin [l])

127 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

--1 (3.66) P l(ch t)=&sh 2 1/2 t F(vl+iv 7l-ill 2,3 -sh 2 t) ill7 and hence from (3.65) one has ZE(t) = F(Y),Y,$ -sh 2 t) as required by (2.2). "n To obtain resolvants R (t,p) for n = 1, 2, . . . we now multiply the Zn(t) of (3.64) by an appropriate weight factor. ll Since one has for example -n --1 n+21 -n- 1 2 'sh(t3) F(r,F,n+-,-shn+l+ip n+l-ip 3 2 t) (3.67) P :(ch t) = 2 ip-7 r(n + $

(cf. Robin [l]),in order to satisfy the resolvant initial conditions (2.3), it is natural to take (cf. (3.21) - (3.22)

h n+l+iu n+l-iu 3 2 (3.68) Rn(t,u) = F(TFn+ph t)

We now use (3.23) - (3.24) to obtain recursion relations for the in.Some routine calculation yields then

Theorem 3.17 The resolvants infor the case of three dimensional Lobazevskij space are defined by (3.68) and satisfy (v = iu-1)

(3.69) ^R! + (2n+l) coth t in= (2n+l) csch t

128 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

h (3.71) R:t +(2n+2) coth t s: + n(n+2)^Rn = v(v+2)^Rn

These equations are in agreement with Silver [l]where they are obtained in a different manner and the resolvants are expressed somewhat differently in terms of the functions Pa of B ,Y Vilenkin [l]. One can now proceed as before to determine resolvants Rn and canonical sequences of singular Cauchy problems.

Similarly one can expand the matrix theory to deal with higher dimensional Lobazevskij spaces (cf. Silver [l])but we will not spell this out here (cf. Section 4). The question of deduc- ing the recursion relations (3.69) - (3.70) from group theoretic information about the irreducible unitary representations of G in the f-l will be deferred for the moment. u

Remark 3.18 Going back to the general format of Sections

2.1 - 2.2, Example 3.16 corresponds to the case ma = 2, mZa = 0, p = 1, da = -p(p+l), dpa = 0, and n = p. Then using the Kumner relation indicated at the end of Section 2.2 with z = thLt now, 2 we have z/z-1 = -sh t and for T - (p,o)

(3.72 Y (a K) -u,T t

ip+l+ iu+2+p 3 2 = c thPt ch-iu-lt F(+, ,p+Z,th t) -uYT

129 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Hence we can write from (3.68)

A (3.73) RP(t,u) = c-l shePt Y (atK) -uYT -u ,'c exactly as in (3.7). Putting this in (2.11) yields (3.71) and the recurrence relations follow as before from (3.23) - (3.24).

2.4 Expressions for general resolvants. Let first ma = m and m2a = 0 so that p = m/2, R = iv + T, da = -p( p+m-1) , and

= T d2a 0, while - (p,o).

Theorem 4.1 Resolvants for the case ma = m and m2a = 0 are given by

A (4.1) RP(t,u) = c-l shwPt Y -u ,f (atK) -uYT R+ R+p+l +m+l 2 = ch-P-R t F(+, ,p 7tht)

These satisfy the resolvant initial conditions as well as the differential equation and sDlittinq recursion relations below.

A A 2 m2"p- (4.2) RtPt + (2p+m) coth t R! + [p(p+m) + p + ($ ]R - 0

A A A (4.4) R! + (2p+m-1) coth t Rp = (2p+m-l)csch t RP-'

130 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

Proof: Equation (4.1) follows from definitions (cf. (3.7) and (3.73) for motivation), (2.9) the Kumner relation F(a,b,c,z)=

(l-z)-aF(a,c-b,c,z/z-l), and an extended version of (3.67); the recursion formulas (4.3) - (4.4) follow from the known relations

(cf. Robin [l]for details). Finally (4.2) results from (2.11) or (4.3) - (4.4). QE D

In the situation now when mZa = 1 the situation becomes somewhat different. We recall that T -+ (p,q) with (p,q) E Z+ x Z and 2 p 2 q E 22,. We take ma = m so that da = -p(p+m) + q and dpa = 2 m For the resolvants we use (3.7) again (cf. -4q with p = -2 + 1. (3.73) also) with (2.9) to obtain

= ch-P-2XF(x+y, x + y,y, th2t)

1 R where x = -(ip + 2 + 1) = and y = p + + 1. An elementary 2 2 2 computation now yields the resolvant equation from (2.11) in the form

(4.8) *R;iq + [(Pp+m+l)coth t + tht]c!7q

2 + [p(p+m+2) + p + (!)+1)2 + q2~ech2t]"R9q= 0

131 SINGULAR AND DEGENERATE CAUCHY PROBEMS

Theorem 4.2 For the case = 1 with % = in resolvants are given’by (4.7) a satisfying the resolvant initial conditions (2.3) and (4.8). There are various spl itting recursion relations which we list below.

(4.9) p - 2x1 tht ^Rp’q

sh2t tht 6p+zq

A (4.10) RPtYq = 2(y-l)coth t sech2t ip-2sq

2(x+-l)(y-x - y-1) + [2(1-6)~0tht + 1 - qlth t] Y-2

(4.11)

A Rp’q = -q (4.12) t th t SPsq -Z(y-l)coth tiPsq

+ Z(y-l)csch t ^Rp-lSq-’

(4.13)

A Rp’q = qth t -Z(y-l)coth t (4.14) t iPsq iPsq

+ E(y-l)csch t ip-l’q+l

132 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

h 2 coth t (4.15) RPtSq = q th t iPsq+ -=x+y) 9+ 1

h 2 coth t (4.16) R!’q = -q th t iPsq+ (X+Y) q-1

(y-x -

Proof: The recursion relations are derived using the for- d mula F(a,b,c,z) = (ab/c)F+ = (ab/c)F(a+l ,b+l ,c+1 ,z) and various contiguity relations for hypergeometric functions (cf. Magnus-Oberhettinger-Soni [l].Thus to obtain (4.9) - (4.10) one uses the easily derived formulas

(4.17) (l-z)F+ = F + w,zF(a+l,b+l,c+2,z)c c+l

(4.18) abz(1-z)F+ = c(c-l)F(a-1 ,b-1 ,c-~,z)

+ c[J-(z - (c-bz-l)]F

For (4.11) - (4.12) one uses the formulas

(4.19) b(1-z)F+ = CF + (b-c)F(a+l ,b,c+l ,z)

(4.20) abz(1-z)F+ = c(c-l)F(a-1 ,b,c-1,z)-c(c-bz-l)F whereas for (4.13) - (4.14) we utilize

(4.21) abz(1-z)F+ = c(c-l)F(a,b-1,c-1 ,z) - c(c-aa-1)F

133 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(4.22) a(1-z)F+ = CF - (c-a)F(a,b+l ,c+1 ,z) and finally for (4.15) - (4.16) one has

(4.23) bZ(l-z)F+ = c(z+~-~+~(b-c))F - F(a+l ,b-1 ,c,z)

(4.24) abz(1-z)F+ = F(a- -bcoa-b-1

Let us also remark briefly about the splitting phenomenon. Thus for example (4.19) yields

I (4.25) F = c(1-z)-a [cF + (b-c)F(a+ whereas (4.20), after an index change gives

I (4.26) F (a+l,b,c+l ,z) = z(l [cF - (c-bz)F(a+l ,b,c+l ,z)]

Now differentiate (4.25), insert (4.26), and then use (4.25) again to eliminate F(a+l,b,c+l,z). Multiplying by z(1-z) one I1 obtains then the hypergeometric equation z(1-z)F + I [c - (a+b+l)z]F - abF = 0. It is easy to show that if the hypergeometric equation splits in this manner then so does (4.8) under the composition of (4.11) - (4.12), for example. QED

For completeness we will write down some formulas for the

cases m = 3 or 7 but will omit the recursion relations since 2a the pattern is exactly as above. Thus for m2a = 3 we set ma = m

and recall that (p,q) E Z+ x Z+ with p ? q E 22,. One has

134 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

m p = -+ 3 and R = iu + p with d2a = -4q(q+2) and da = -p(p+m+2) + 2 h q(q+2). We use (3.7) and (3.73) again to define Rpyq and from (2.9) this yields

1 m where x = R/2 = $iu +%+ 3) and y = p + -i- + 2. The differential equation arising from (2.11) is then

* (4.28) R!tq + [(2p+m+3)coth t + 3 th t]i!yq

= [p(p+m+6) + p2 + (!=)+3) 2 + q(q+2)sech2t]RPyqA = 0

For mZa = 7 one has p = !+2 7 and setting ma = m it follows that da = -p(p+m+6) + q(qt6) with dZa = -4q(q+6). In this case

from (3.7) and (2.9) again

h R+ Rtp-q-6 2 (4.29) RPYq(t,u) = ch-P-a t F(9,+ ,p+%+4, th t)

* and from (2.11) the differential equation for RPyq is

h (4.30) R;tq + [(Ep+m+7)coth t + 7th t]6!’q

2m 2 + [p(p+m+l4) +p + (9+7) +q(q+6)se~h~t]~~’~= 0

Theorem 4.3 For m2a = 3 (resp. 7) the resolvants are given

by (4.27) (resp. (4.29)) and satisfy (4.28) (resp. 4.30)).

135 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

These results completely solve the rank 1 case and the con-

nection of the Fourier theory to the associated singular partial

differential equations has been indicated already (cf. Theorem

3.13 and Chapter 1).

2.5 The Euclidean case plus generalizations. We follow

here Carroll-Silver [15; 16; 171 and will indicate results only

for one simple Euclidean case; a complete exposition appears in

Silver [l]. Thus let G = R2xySO(2) and K = SO(2) where (;,a)

(;,8) = (&y(ct)?,ct+B) is a semidirect product with

(cf. Helgason [l],Miller [l; 21, and Vilenkin [l]) for background information). Thus G = M(2) is the group of orientation and distance preserving motions of R2 and is a split extension of R2 (with additive structure) by K so that R 2 and K are sub- groups of G with K compact and R2 normal (cf. Rotman [l]).

Elements g = (;,a) can be represented in the form

'cos ct -sin ct xl' (5.2) g = sin ct cos ct x2 ,o 0 1,

and multiplication is faithful. As generators of the Lie alge- ." bra g of G we take

136 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

00 000 0 -1 0' (5.3) al = 1 0 l;i a2 1; ; b;; a3 = j100 00 0 0 0, with multiplication table

." Thus g is solvable (but not nilpotent, nor semisimple). Now set

V = G/K and since (;,a)(;,@) = (z,a+B) there is an obvious global 2 analytic diffeomorphism V -+ R . One defines as before L(g)f(x) = f(g-%) = f(y(-a)(;-;)) where g = (;,a) and g-l = (y(-a)(-;),-a).

One thinks here of ;; as r(h) = IT(;,@) = IT((;,O)(O,@)) so that g-'; = g-lr(h) = IT(g-'h) = r(y(-a)(-;) + y(-a);,@-a) = y(-a)(;-G).

." If f is differentiable then L induces a representation of g as in x1 (3.29). Writing Ai = L(ai) we have first, for $ = ( ) = -+ x2 +. r(x,@) = (xl,x2) (for simplicity of notation), exp(-tal)x = +. (xl-t,x2), exp(-ta2)z = (xl,x2-t), and exp (-ta3); = y(-t)x.

Consequently we obtain A1 = -a/axl, A2 = -a/ax2, and A3 = x2 a/axl-xl a/ax2. Writing H+ = A1 + iA2, H - = A1 - iA2, and H = 2iA3 it follows from (5.4) that (note the contrast here with

(3.12) and (3.26))

(5.5) [H,H+] = 2H+; [H,H-] = -2H ; [H+,H-] = 0

The Riemannian structure on V is described in (geodesic) polar 22 coordinates by ds2 = dt2 + t d8 (cf. Helgason [l]) and the geometry is "trivial." There follows immediately (cf. Vilenkin

137 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Lemma 5.1 In polar coordinates (t,e) on V

it3 (5.6) H+ = -e H = -2i [a/at + Lt a/ael; a/ae;

-i0 i H- = -e [a/at - a/ael

221 122 (5.7) A = H H = a /at + - a/at + a t /ae +- t

As before (cf. Proposition 3.7) we work with (dense) differentiable basis "vectors" fi in Hilbert spaces tl where G provides u a unitary irreducible representation and these are character- L u ized by the conditions (cf. Miller [l;23, Vilenkin [I])

= iuf;+l; H-f; = ipfi-l; H3f; = mfi where H3 = 1/2 H and 1-1 is real (see Carroll-Silver [15; 16; 171 and Silver [l] for further details). Here we can write L = (1.5) ) , but emphasize that the semi simp1 e theory does not apply. There results (cf. Vilenkin [l])

Theroem 5.2 Canonical basis vectors in H can be written u in the form

(5.9) fi = (-i)mexp (ime)Jm(pt)

Proof: Take fi = eimOwL(t) which will assure the third

requirement in (5.8). Then the wi must satisfy

138 2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

(cf. (5.7 - (5.8)). The solutions J,(ut) are chosen for finite- ness cond tions and the factor (-i)mmakes valid the recursion relations indicated in (5.8) (cf. Vilenkin [l]); the resolvants age given by "R(1-I,t)= (i)m2mr(m+l)(pt)-mwi(t)and satisfy (1.3.11) - (1.3.12) with A(y) replaced by u 2 (resp. y by 1-11 - similarly (1.3.4) - (1.3.5) hold for ^Rm with ? = 1. QED

Remark 5.3 It is easy to show that the mean value as defined by (1.6) coincides in this case with the mean value uX(t) (cf. formula (1.2.1) - and see Carroll-Silver [16] for details). Thus the results of Chapter 1 may be carried over to this group theoretic situation , which i s equivalent.

We conclude this chapter with some "generalizations" of the growth and convexity theorems (1.4.12) and (1.4.16) in the special case V = SL(2,R) /SO(2) (cf. Theorem 3.13 and Carroll- Silver [15; 16; 173). First it is clear that if f _> 0 then, referring to (3.43), (Mt # f)(v) = (Mtf)(v) 2 0 for m 2 0 an integer (here we assume f E C'(V) for convenience later). Hence um(t,v,f) = um(t,v) 2 0 when f 2 0 by (3.44) for m > 0 an integer. Now write Am = A - m(m+l) so that by 3.45) we have recall Mt = a M t,

Theorem 5.4 Let A,f L 0, m 2 0 an integer; then um(t,v) is monotone nondecreasing in t for t 2 0.

139 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Proof: AM f = M (Af) by (1.7) and Lemma 1.1. QED rl rl

Now let $ be any function such that d$/dt = csch2m+1t so that d/d$ = sh2m+1t d/dt (cf. Weinstein [12]). Then (3.47) can be written

(5.11) a 2 /a$ 2mu (t,v,f) = ~h~~~tum(t,v,A,f)

Consequently there fol lows

Theorem 5.5 If A,,,f L 0 then um(t,v,f) is a convex function of $.

Remark 5.6 Working in a harmonic space Hm(cf. Ruse-Walker-

Willmore [l]), Fusaro [l]proves (M = M(v,t,f) as in Theorem

3.11)

(5.12) Mtt + (w+ log' g(t)'l2)Mt = AM where g = det (9. .), gij denoting the metric tensor, and g depends 1J on the geodesic distance t alone (A denotes the Laplace-Beltrami operator). If Af 2 0, M will be nondecreasing in t and a convex function of $ where $'(t) = t'-m/g(t)'/2. Weinstein 1123 works 2 in spaces of constant negative curvature -a (which are harmonic) and proves similar theorems for

(5.13) Mtt + a(m-1) coth (at)Mt = AM

We refer to Helgason [4] for such "Darboux" equations in a group context; the extension to "canonical sequences" is due to Carroll

140 2. CANONICAL SEQUENCES OF SINGUALR CAUCHY PROBLEMS

[21; 221, Carroll-Silver [15; 16; 171, and Silver 113.

141 This page intentionally left blank Chapter 3

Degenerate Equations with Operator Coefficients

3.1 Introduction. In the preceding chapters we were concerned with Cauchy problems for evolution equations with operator coefficients that were permitted to become infinite or singular. We turn now to the dual situation in which the operator coefficients are permitted to degenerate in some sense. The examples to follow will illustrate some typical partial differential equations that occur in the initial boundary. value problems to which our abstract results will apply.

Example 1.1 Various diffusion and fluid flow models lead to the partial differential equation

with nonnegative real coefficients. This equation can be elliptic (ml = m2 = 0), parabolic (ml > 0, m2 = 0), or of pseudoparabolic or Sobolev type (m2 > 0), and the type may change with position = x and time = t. The equation (1.1) with m2 > 0 was proposed in 1926 by Milne [l]. Similar equations have been proposed for numerous other applications by Barenblat-Zheltov- Kochina [l] , Benjamin [l], Benjamin-Bona-Mahoney [2], Buckmaster- Nachman-Ting [l], Chen-Gurtin [l], Coleman-No11 [1], Huilgol [l], Lighthill [l], Peregrine [l; 21, Taylor [l], and Ting [2]. This equation appears below in Examples 3.5, 4.5, 4.6, 4.7, 5.10,

143 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

5.13, 5.14, 5.15, 6.21 and 7.6.

Example 1.2 A partial differential equation analogous to

(1.1) but containing a second order time derivative was introduced in 1885 by Poincarg. Similar equations of the form

= f(x,t) with nonnegative coefficients have been proposed; here An is the

Laplacian in x E Rn and An,1 denotes the first n-1 terms of An.

Applications of (1.2) are given by Boussinesq [l],Lighthill [l],

Love [l],and Sobolev [l]. Equations of the form (1.2) are considered below as Examples 3.9, 4.8 and 6.22 (cf., Theorem 3.8).

Remark 1.3 The name SoboZev equation has been used extensively to designate partial differential equations or, more generally, evolution equations with a nontrivial operator acting on the highest order time derivative. This operator is usually--not always--el liptic inthe space variable.

Example 1.4 Certain applications lead to problems with standard evolution or stationary equations in a region G of Rn but with a constraint in the form of a partial differential equation on a lower dimensional submanifold (e.g., the boundary) of

G. Thus one may seek a solution of

144 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

DtU(xyt) - AnU(X,t) = f(x,t), x E G, (1.3)

Dtu(syt) + au/aN = div(a(x) grad u(s,t)), s E S

where S is a n-1 dimensional submanifold, N denotes a normal,

and the divergence and gradient are given in local coordinates

on S. Such a problem was given by Cannon-Meyer [l]to describe

a diffusion process which was "singular" on S (cf. Example 6.21).

Example 1.5 Problems similar to the above but of second

order in time occur in the form

-Anu(x,t) = f(x,t), x E G,

(1.4) 2 Dtu(syt) + au/aN = 0, s E aG

to describe gravity waves (cf. Whitham [l]).One can view this

as a degenerate problem in which the coefficient of utt is the

boundary trace operator. It can be handled as the preceding

(cf. Showalter [5]) or directly as by Friedman-Shinbrot [S] or Lions [lo], Ch. 1.11.

Example 1.6 Degenerate parabolic equations arise in various

forms other than (1.1). Problems from mathematical genetics

lead to

where the leading operator degenerates at t = 0 and the second

145 SINGULAR AND DEGENERATE CAUCHY PROBLEMS degenerates on the boundary of the interval [O,l] (cf. Brezis-

Rosenkratz-Singer [2], Friedman-Schuss [l],Kimura-Ohta [l],

Levikson-Schuss [l],and Schuss [l;23 for (1.5) and related problems). Also, (1.5) is considered below in Example 5.15. De- generacies can occur in parabolic problems as the result of non- linearities. This is the situation for the equation

of flow through porous media (cf. Aronson [l]and Example 6.23).

Example 1.7 Wave equations occur with degeneracies, typically either in the form of (1.2) with cl(x,t) ? 0 and c2 z 0 or in the form of (1.1.4). The latter situation will be discussed in

Section 3.2 below.

Each of the preceding examples is a realization in an appropriate function space of one of the abstract Sobolev equations

(1.7) ~(M(u))d + L(u) = f

for certain choices of the operators. These two equations will be considered in various forms in this chapter. In Section 3.2 we use spectral and energy methods to discuss (1.8) when C is the identity and B and A are (possibly degenerate) operator polynomials involving a closed densely defined self adjoint operator.

This covers the situation of (1.1.4). The equation (1.7) is

146 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS called strongly regular when M is invertible and M-lL is continuous on some space. Similarly, (1.8) is strongly regular when

C-lB and C-lA are continuous. The strongly regular case is studied in Section 3.3; the operators are permitted to be time- dependent and nonlinear. We call the equations weakly regular when their leading operators are only invertible. We shall give well -posedness results in Section 3.4 for 1inear weakly regular equations by means of the classical generation theory for linear semigroups in Hi1bert space. Section 3.5 treats degenerate linear time-dependent equations by the energy methods of Lions

[5]. Applications include Examples 1.1, 1.4, 1.6, second order evolution equations of parabolic type (cf. Theorem 5.9), and many others (cf. Showalter [Z]). Related nonlinear problems are discussed in Sections 3.6 and 3.7 by methods of monotone nonlinear operators. These techniques also allow the treatment of related variational inequalities (cf. Theorems 3.12 and 6.6). Each section contains a list of references to related work. The partition of these references is inexact, so one should check all sections for completeness.

3.2 The Cauchy problem by spectral techniques. Referring to Remarks 1.1.3 and 1.1.4 we will deal first with an abstract version of the degenerate Cauchy problem in the hyperbolic region with data given on the parabolic line, following Carroll-Wang

[12] (cf. also for example Berezin [l],Bers [l],Carroll [7; 9;

341, Conti [3], Frank'l [l],Krasnov [l;2; 31, Lacomblex [l],

147 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Protter [2], Walker [l; 2; 31, Wang [l;21 as well as other pre- vibus citations in Remark 1.1.4). Thus consider (I denotes d/dt)

II (2.1) U + AaS(t)ul + A'R(t)u + Aq(1)u = f where A is a closed densely defined self adjoint operator in a separable Hilbert space H with (Ah,h) t cllh112, c > 0, a,@ 2 0,

1 = A-' E L(H), q(1) = a(t) + B(t)l where a(t) vanishes as t -+ 0

(B(t) E L(H), S(t) E L(H), and R(t) E L(H)) while f is "suitable"

(see below). It is assumed that all operators comnute and we 2 I seek u(*) E C (H) satisfying (2.1) with u(o) = u (0) = 0 (other initial conditions can also be treated but we omit this here - cf. however (2.32)). We will use first the technique of Banach algebras and spectral methods developed in CarrollE4; 6; 8; 10;

11; 14; 29; 30; 311, Carroll-Wang [12], and Carroll-Neuwirth [32;

331; cf. also references there to Arens [l],Arens-Calderh [l], V Dixmier [l],Foias [3], Gelfand-Raikov-Silov [8], Lions [5; 61,

Rickart [l],and Waelbroeck [2]. We obtain results "similar" to those of Krasnov [l;2; 31 and Protter [2] but in a more general

operator theoretical frameword; moreover in our development a( 0) need not be monotone and some new features arise (see Remarks 2.10 - 2.12 and cf. also Theorems 2.16 and 2.17).

Let us assume for i lustrative purposes that S(t) = Ss(t),

R(t) = Rr(t), and B(t) = Bb(t) where B, R, S, and 1 commute in L(H) and are normal; hen e by a result of Fuglede [l](l,B,R,S, *** B ,R ,S ) are a commuting family and we denote by A the uniformly

148 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS closed * algebra generated by this family and the identity I.

Note that for any h E H, ABlh = AlBh = Bh automatically for example, and the comnutativity of say A with B means that for h E D(A), Bh E D(A) with ABh = BAh. It will be assumed that b, 1 r, and s belong to Co[o,T] and a E C [o,T] where T < m.

Remark 2.1 A few remarks about finitely generated commutative Banach algebras A with identity will perhaps be useful in what follows (cf. Carroll [14] and references there). Let + :

A + $ be a continuous homomorphism so that ker + is a (closed) maximal ideal in A with A/ker + Z 6 and one identifies I$ with ker +. The set QA (called the carrier space) of maximal ideals can then be identified with a weakly closed subset of the unit

I A A ball in A and we write +(a) = a(+) for a E A. Since I(+)= 1, if A is a commutative * algebra in L(H) with generators ao, * * al, . . ., an, al, . . ., an, and I (ao self adjoint), one can show that QA is homeomorphic to the (compact) joint spectrum A A "* UA = {a(+)>= {(ao(+), al(+), . . ., an(+))> C $2nt1 where A* A A a,(+) = ak(O) and a,(+) is real. The functions ak will then be continuous on QA - uA and A will be isometrically isomorphic to C(QA) where C(QA) has the uniform norm. In our situation above 7 we have uAC$ and we associate the complex variables (z 0' ' *' A A A* z,) to (I(+),B(+), . . .,S (+)) with h = l/zo 2 c and lzkl 5 c1 = max(IIBII,IIRII,IISII) for k 2 1. One notes that zo -+ 0 corresponds to X -+ m and we will call the map r : + a : C(QA) =

C(U,) -+ A the Gelfand map. A further notion of use here (cf. also

149 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Section 1.5) is that of decomposing H by an isometric isomorphism 0 : H -+ H = I@H(z)dv which diagonalizes A and r may be extended, as an isometricuA isomorphism, for example to all bounded

II Baire functions B(uA) with values in the von Neumann algebra A

(here A’lC L(H is the bicommutant of A and is the closure of A in say Ls(H)) we will use uA for convenience in measure theoretic arguments instead of QA. The so called basic measure v arises naturally (cf. Dixmier [l]or Maurin [l; 31) and we will not give details here. A v-measurable family of Hilbert spaces H(z) on uA consists of a collection of functions z -+ h(z)

E H(z) such that there is a vector subspace FCIIH(z) with z -+

1) h(z) Ilff(z)measurable (h E F), (h(*), g(-)) measurable for H(z) all h E F implies g E F, and there is a fundamental sequence hn E

F such that the closed subspace of H(z) generated by hn(z) is 2 H(z). Now under the present circumstances ff = L,,(uA,ff(z)) is a - .. Hilbert space with norm [I Ilh(~)1)~dv]~’~= llhllH and a Lebesque 0- dominated convergence theorh will be valid. Diagonalizable operators G E L(H) are defined in the obvious manner (cf. Section

1.5) with G = 0-lG0 E L(H) where G - G(z) E L(H(z)), all arguments being carried out in Ls(H) for example.

Now in connection with (2.1) we consider the equation (under our assumptions S(t) = s(t)S, etc.), obtained by applying 0 to

(2.1)

Al I ni n n (2.2) u + haz3s(t)u + hbz2r(t)u + h[a(t)+zozlb(t)]u = 0

150 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS where ^u = eu and for example We write z = (z - z3' 1' ' - ') z6) and by Coddington-Levinson [l]or Dieudonn; [l](cf. also

Chapter 1) there exist unique solutions Z(t,-r,z,A) and Y(t,-c,z,A) of (2.2) such that Z(T,T,Z,X) = 1, Z (T,T,z,A) = 0, Y(T,T,z,X) = t

0, and Yt(TyTyzyX) = 1 for 0 5 T S t 5 T < 00, lzkl 5 cl, and c 5 X < a. The functions Z and Y will be continuous in (t,T,z,h) in this region and we need only check their behavior as A * a.

To this end we construct a "Green's" matrix as in Sections 1.3 and 1.5 (cf. (1.3.13)) in the form

which will satisfy the equation

(2.4) ~A(tyTyzYA)a + A1~2HA(tyZ,h)GA(t,~~ZyX)= 0

(2.5) HA( Y Y A) = a- 1 1a(t)+zozl b(t)+AB-lzpr(t) X 'z3s(-l t)1 As in Chapter 1 it follows that (cf. (1.3.15))

(2.6) ea(t,T,Z,A)a -X1/ZGA(ty~,z,X)H (T,z,X) = 0 A A

A A * Setting u1 = u and u2 = X-'/2c' with $ ) we obtain formally from zt + ~'/2HA(t,z,~)ii= -i the solution of (2.1) in the form

151 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

f = ef. In order to check the behavior of Y we replace t by 5 in (2.2) for Y and multiply by (<,-c,z,X); upon taking real parts v5 and assuming now that a(t) is real valued one obtains

2 28 2 Now note that IrhBYy I 5 lYI2 + IY I ) so that, upon 5 (Irl X 5 integration from T to t,

Assume now that Re(z3s(<)) 2 0 if a > 0 with 28 <, 1 and take X >-

1 so that X2' 5 X (recall X >, c and if c < 1 a further argument will apply - see below). If = 0 then the A" term can be in- corporated into the right hand side of (2.9) or (2.10) in an obvious manner. We have then (since lzkl 5 c1 for k 2 1)

152 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

t (2.10) (YtI2 + Aa(.t)IYI2 5 1 + 2cl 1YS12dS T

It Aa(S)IY12dS to the r ght hand side of 2.10) and using Gronwall's lemma (Lemma 1.5.10) one obtains, sett ng

E(t,S) = exp 2cl(t-S),

t (2.12) IYtI2 + Aa(t)lYI2 5 E(t,r) + I AP(S)E(t,S)lY12dt T

The following lemma now gives a somewhat sharper estimate on 2 IYI than can be obtained by a direct application of the Gronwall lemma; the proof however is a simple variation.

Lemma 2.2 Given (2.12) with P 2 0 and a > 0 it follows that for 0 < T 5 t 5 T < m and A ' 1

(2.13) Aa(t)lYI2 5 E(t,T) exp (1: # da)

Proof: We first omit the term IYtI2 on the left hand side t of (2.12) and set X(t,T) = [ AP(S)E(t,S)lY12dg so that X+(t,r) = L t '9 AP(t)lY12 + I AP(S)Et(t,S)IYI dS = AP(t)lY12 + 2clX(t,T). Then r - multiplying (i.12) (with lYtlZ deleted) by P(t) and using the last relation for Xt we obtain

(2.14) .a(Xt-2clX) 5 PE + PX

153 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Thus defining F(t,T) = exp (- 2cl]dS) for T > 0 we have from (2.14), (FX)t 2 (P/a)EF. But E(t,T)F(t,T) = exp(-/:[:]dt) and hence (FX)t 5 -(e~p(-l:[:]dS))~ from which follows

since F(T,T)X(T,T) = 0. This may be written

which proves the lemna QED

c1 21 We observe from (2.11) that P/a = a'/a + (,)[lrl +xlb12] so that t (2.17) exp (I [:Id<) = #exp (jtcl[&+a &Id<)Aa T T If c < 1 we can carry through the estimates with lrlz replaced by I r I 2A28-1 when A < 1 (recall 28 5 1 ) and from Lemma 2.2 we obtain, using (2.17),

Lema 2.3 Given (2.12) with P 2 0 and a(-c) > 0 for T > 0 we have for 28 5 1

." = = where c1 c 1 max (1, c"-l) and c2 exp 2clT.

Define now +(t,.) = exp c a and $(t,T,A) = ..d ..d - 1' 'T " T -F of exp - -kfa dg and as 0 these functions may course AT

154 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS become infinite. Noting that $(t,T) 5 $(T,T) and $(t,T,A) 5

$(T,T,c) we can state

Corollary 2.4 Let $(T) = $-’(T,T) and $(T) = $-’(T,T,c); then

where Y = Y(t,.r,z,A) is the unique solution of (2.2) with P 2 0, a(.) > 0 for T > 0, 28 5 1, A 2 c, and Re(z3s(<)) 2 0 satisfying

Y(T,T,z,A) = 0 with Yt(~,~,zYA) = 1. We set now W(t,T,z,A) = Q(T) Y(t,-c,’z,A) where Q = ($$a) 1/2 and observe that the estimate (2.19) holds for T = 0 also while

W is continuous in (t,T,z,A) (Q(T) + 0 as T -+ 0). If h E H then

then (cf. Remark 2.1) (t,.r) -+ W(t,T) = 0-lWB :. h + A1/‘Wh is 1/2 2 continuous since I IlA W0hlI dv will converge appropriately by aA Lebesque dominated convergence for example (in fact A’/’W E B(aA)

II can be demonstrated, so r(A1/2W) E A but this will not be needed - note here that All2 and W also conute on uA). One can now prove

Theorem 2.5 Under the assumptions of Coro lary 2.4 (and

Lemma 2.3) we have, with y = max (l/Z,cx) and 0 5 T 5 t 5 T < m,

155 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Proof: We need only check the bounds since the rest will follow from the Lebesque dominated convergence theorem. From (2.12) and (2.16) - (2.17) one has

(2.21) IYtI2 5 # E(t,Y)$(t,T)$(t,T,x) 2 SO that = c3 where c = c2 max a(t) (max on lWtl IQYtI2 5 3

[o,T]); actually (t,.r) -f Wt(t,.r) E Co(Ls(H)) since Q(T) -t 0 as

T + 0 and this is useful later. For the last estimate we go back to (2.2) to obtain the inequality (recall that 2f3 5 1)

5 c4xa + c5x 1/2

follows. We again observe that (t,.r) -t Wtt(t,T) E

QED

Now we consider (2.7) and will show that it represents a

solution of the Cauchy problem (2.1) when f E Co(D(Ay)). Thus

setting h(c) = f(<)/Q(c) (2.7) may be written in the form t u(t) = W(t,<)h(c)d< and we can consider this as a Riemann type 0 (vector valued) integral (in order to carry the closed operators

Aa, A', and A under the integral signs). The following computa-

tions are then justified by Theorem 2.5 and remarks in its proof.

First u(o) = 0 and

156 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

1 Theorem 2.6 Assume a E C [o,T], a(t) > 0 real for t > 0, a(o) = 0, by r, s E Co[o,T], 28 5 1, y = max(a,l/2), Re(z3s(t)) 2

0, P 2 0, and f/QE Co(D(Ay)). Then there exists a solution of 2 (2.1) given by (2.7) with u(o) = ut(o) = 0, u E C (H), u E

C'(D(Ay)), and u E C0(D(Ay+'I2)).

For uniqueness we will give several results. First from

(2.6) one obtains (cf. 1.3.17)

(2.25) Y = -Z+A"Z~S(T)Y T and some bounds for Z must be established. Duplicating our estimates leading to (2.9) we have

t (2.26) IZtI2 + 2Re(Xaz3s(S T

and under the same assumptions and procedures as before it follows that (cf. (2.9) - (2.12)) t (2.27) Aa(t)lZ12 + lZtL2 5 Aa(-r) + 2c 1 1Z512dE lT

157 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Then by a version of Lemmas 2.2 and 2.3 one obtains

Consequently we have proved

Lemna 2.7 Under the hypotheses of Corollary 2.4,

@(T)vJ(T)lZl25 c2.

We set q = (@vJ)”‘ with V(t,T) = e-1q(T)Z(t,T,z,A)8 so that by Lebesque dominated convergence again V(t,T) E L(H) (and II V(t,T) E A can be shown but again this is not needed). Next we observe from (2.6) again that

a- 7 so that IQ(T)Z~~5 c7A1l2 while from (2.25) lYTl 5 c& for .“ a 5 1/2 with lY,l 5 c6 for a 2 1/2. The case a 5 1/2 is essentially trivial and will not be discussed further. Thus using

Lebesque dominated convergence again we can state that for T > 0, 1 a- 1 T +. Y(t,.r) = e”Ye E C (Ls(D(A z),H)) while T + Z(t,r) = 1 1/2 e-lZ0 E C (Ls(D(A ),H)). Therefore if u is a solution of (2.1) with data prescribed at T > 0 we operate on (2.1) with Y(t,C)

(changing t to in (2.1)) and integrate to obtain for u E 1 5 y t -. Co(D(A ’I), u E C’(D(Ay)), and u E C2(H)

158 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

JT JT

Our hypotheses and lemmas insure that everything makes sense and

using (2.25) with (2.30) plus another integration by parts there

results from (2.31) (2.32) u(t) = Z(t,-clu(r) + Y(tyT)ut(T) + i: Y(t,S)f(S)dS As in Section 1.5 for example we are making use in (2.31) and

(2.32) of the hypocontinui ty of separately continuous maps E x

F +. G when F is barreled. Now as T +. 0, by hypocontinuity again we have (note that this is stated badly in Carroll-Wang [12])

Theorem 2.8 Assume the hypotheses of Theorem 2.6 with con- I

tinuous u/q + 0 and u /Q -f 0 as T -f 0. Then the solution of

(2.1) given by Theorem 2.6 is unique.

In the event that q(t) > 0 for 0 5 t 2 T a somewhat stronger uniqueness theorem for (2.1) can be proved (cf. Carroll-Wang

I 1121). Assume u(o) = u (0) = f = 0, with u satisfying the conclusions of Theorem 2.6, and then we define the operator in

L(H) by t (2.33) L(t,.r) = 0-l exp(-Aaz3 1 s(S)d<)e T Operating on (2.1) with L(t,S), where t has been changed to 5 in

159 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(2.1) we obtain

t (2.34) u’ (t) = -1 L(t,E)CABRr(E)+Aa(E)+Bb(S)lu(.C)dE 0

Since IlAull and IIABuII are bounded in Theorem 2.6 we have by a we1 1 known inequality

(recall that q > 0 means lr21a dC < w and hd.!a dc < a). NOW 0 0 Z(t,T)u(T) -+ 0 automatically in (2.32) as T + 0 since qZ is bounded and u + 0 while the term Y(t3?)ut(T) can be written

’0 where 6 < 1/2. But Q = qa1j2 so q > 0 implies a(T)’I2Y(t,T) E Ls(H,D(A 1/2 )) by Corollary 2.4 and heorem 2.5 while by (2.35) u’(.)/(radE)6 -+ 0 as T -+ 0. Hence we have 0

Theorem 2.9 Let u be a solut on of (2.1) satisfying the conditions of Theorem 2.6 with q(t) >OforO

(ra(E)dE)6/a(r)’/2 bounded (6 < 1/2). Then u is unique.

160 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Remark 2.10 We emphasize here that a(-) is not required to be monotone, in contrast to Protter [2] or Krasnov [l],since by I 2 (2.111, P(t) 2 0 for all x merely implies a 2 -cllrl (examples of nonmonotone a are given in Carroll-Wang 1123 and Wang [l;23).

The condition of Protter [2] (cf. also Berezin 111, Bers [l]),

phrased here in the form tr(t)/a(t)’/’ -+ 0 as t -+ 0 (with a(-) monotone), for solutions of a numerical version of (2.1) to be well posed in a local uniform metric, has its analogue here in 22 the conditions on +(t). Thus for example if A -. -3 /ax , a(t) = tm,AB R(t)u = r(t)ux = tnux, and A”S(t) = s(x,t) with n+l~ A 1 B(t) - b(t) then it follows that tr(t)/a(t)”‘ = t -+ T 0 if n > m - 1 whereas j;lr,‘/a d< = 1: 1: 2n-mdS = (1/2n-m+l) (?n-m+l t2n-ni+l 2n -m+l ) so that +(t) = k exp (c,t /Zn-m+l) > 0 m1 2 when n > The L condition of Krasnov [l]involvesmonotone T - -.2 -- 1 a(t) = O(tm) and r(t) = O(t2 y(t)) where y(t) * 0 as t -+ 0 but

Krasnov is dealing with weak solutions. Now our existence and uniqueness conditions are phrased in terms of u/q, u-/Q, and f/Q 2 where q2 = +$ and Q = $$a which permits a rather precise com- I parison between the behavior of f, u, and u as t + 0. The 9 term seems somewhat curious however since for example if

h Ibl 2 k < m then 1: Ibl /a de 2 (i2/-m+l)(T -m+l-t-m+’) for a(t) = tmso that $(t) = 0 (exp k t-mtl/-m+l) + 0. for m > 1 which im- 1 poses growth conditions on b(t) in order that $(t) > 0. This seems to indicate that the role of the i) term should be investi- gated further and some comnents on this are made in Remark 2.12.

161 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Walker, in as yet unpublished work, investigates by spectral methods (using Riesz operators Rk defined by a/axk = i(-A) 1/2 Rk) the question of directional oscillations and well posedness (cf. also Walker 11; 2; 31)

Remark 2.11 In Wang [2] it is pointed out that if a(-) is monotone near t = 0 (locally monotone) then (2.35) can be replaced by Ilul(t)lI 5 c10t1/Za(t)1/2 for small t so that in (2.36) one could write Y(t,T)ul(T) = a1/2(T)Y(t,.r)-l/2 *with a (TI ul(s)/a1/2(T) +. 0 as T +. 0. This produces a somewhat stronger version of Theorem 2.9 when a(-) is locally monotone. Examples are given in Wang [2] to show that A(t) = a(S)dS/a(t) may not even be bounded for nonmonotone a(t) = O(tm).

Remark 2.12 First we assume z1 and b(t) are real so that

(2.9) becomes, after elimination of the A" term as before, and adding a term clX

I h Now assume Ib (t)/b(t)l 5 c3 on [o,T] and set c = max(cl,c3);

162 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS then, if zlb(S) ? 0 one can write (for 2f3 5 1 and A 2 1)

A A Setting E(t,<) = exp c(t-C) we obtain from Gronwall's lemma again

(cf. (2.12))

tA A (2.41) E(t) 5 k(t,T) + A/ P(C)E(t,S)jY12dC T * Assuming P 2 0 Lemma 2.2 applies to (2.41) (upon omitting the term lYtl 2 + zlb(t)lY/2 in B(t)) to yield

and as in Lemna 2.3 one obtains (cf. (2.17))

I A Lemna 2.13 Assume Ib /bl 5 c3 with c = max(cl,c3) while

A A Y P 2 0, a 2 0, and zlb 2 0. Then for c4 = exp cT and c1 = max( 1 ,c 28-1 )

Corollary 2.14 Defining $(t,T) and $(T) as in Corollary 2.4 the hypotheses of Lemma 2.13 imply

163 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Thus it is possible to eliminate the JI term in Q and q (cf.

Theorems 2.5, 2.6, and 2.8, Remarks 2.10 and 2.11, and Lemma 2.7) if one assumes zlb 10with Ibl/bl <, c3 in addition to a' 2

-cllrl' (cf. Remark 2.10). This means locally (i.e., near t = T) I that if z1 2 0 then b L 0 with either b 1 0 and 0 5 b(t) 5

I b(T) exp c3(t--r) or b 5 0 with b(t) F b(-r) exp (-c3(t-~)); I suitable oscillations in the sign of b are of course allowed.

On the other hand if z1 5 0 then b 5 0 with I bl = -b so either b' L 0 with 0 L b(t) 2 b(T) exp (-c3(t--r)) or b' i 0 with b(t) 5 b(-r) exp c3(t--r). We refer to Carroll [34] for further remarks.

Remark 2.15 In Carroll [7; 91 some weak degenerate problems are solved using Lions type energy methods. The notation is chosen here to conform to these articles rather than to earlier parts of this section and most technical details will be omitted.

Thus let Q > 0 be a numerical function in Co(o,T], T < m; with I)

increasing as t -+ 0 (a priori Q need not approach infinity but 1 it usually will). Let q > 0 belong to C (o,T] with q(t) -+ 0 as t -+ 0. Let V c H be Hilbert spaces (H separab e for simplicity),

V dense in H with a finer topology, and a(t,-, ) a family of continuous sesquilinear forms on V x V with a(t,v,u) = 'm 2 1 and a(t,u,u) ,> allullv. Assume t -+ a(t,u,v) E C [o,T] for (u,v) fixed and let t -+ B(t) E C'(Ls(H)) on [o,T] be a family of

Hermitian operators. Let w > 0 be a numerical function to be

determined such that w(t) -+ 00 as t -+ 0. Let Fs be the Hilbert

164 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS space of functions u on [o,s], s 5 T to be determined, such that

1.2 2 U(O) = 0, Jlu E L (H), and wu E L (V) with norm = I2 FT $u I H )dt; all derivatives are taken in D (H) (cf. Carroll [14] or Schwartz [5]). Let HS be the space of functions h.. satisfying h(s) = 0, h/$ E L2( H), h'/$ E L2( H), and qh/w E LL(V). We define

... S II (2.45) ES(u,h) = I {qa(t,u,h) + (B(t)u',h) - (u ,h ))dt; 0 ... fS Ls(h) = -(f,h)dt JO 2 where f is given with Jlf E L (H) (here ( , (resp. (( , 1)) denotes the scalar product ...in H (resp.... V)). The first problem is to find u E Fs such that Es(u,h) = Ls(h) for all h E Hs. Then since q > 0 on say [s/2,T] one can apply standard techniques for nondegenerate problems ...(cf. Lions... [5]) to proceed stepwi se and find u E FT satisfying ET(u,h) = LT(h) for all h E HT. After a series of technical lemmas such a u E FT can be found using the

Lions projection theorem (see Carroll [9] for details). Another series of technical lemnas will yield uniqueness and one can state

Theorem 2.16 Assume the conditions above with q(t) 2 t t (( dV$2(<))'-E(0 < E < 1) while Q = :$ (q'VJ2/q)I dW2(E;) 0 o... exists. Then there exists a unique solution u E FT of ET(u,h) = ... 2 L (h) for all h E HT, based on a function w L (w E Co(o,T]). T 4

165 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

We remark that w measures the rate of how rapidly u(t) + 0 as t + 0 but its precise description is somewhat complicated

(cf. below). Another theorem based on energy methods can be obtained as follows. We define K- to be the space of functions 5 rt 1 k(t) = @hdS for h E Hs with suitable E C [o,s] where @ > 0 JO on (o,s] while @(t)+ 0 as t + 0. For suitable choice of the numerical function 6 > 0 with b(t) + m as t + 0 we put a prehil-

(We note that if v = @/q then our w above can be taken to be I 2-E1 w2 = v /v (0 < E, < l).)For suitable $,6as above (e.g., @ = j:d

Theorem 2.17 Under the hypotheses of Theorem 2.16 there n " .., exists a unique solution u E KT satisfying ET(u,h) = LT(h) for 1 all h E HT such that IlullA 5 c(/ l$flidt)1/2 KT 0

3.3 Strongly regular equations. We consider first the nonlinear equation

I (3.1) Fl(t)u (t) = f(t,u(t))

in a separable and reflexive Banach space V. For each t E

Ia: [o,a] we are given a continuous linear operator M(t) E

I L(V,V ). Denote by Bb(uo) the closed ball in V centered at uo with radius b > 0, and suppose we are given a function f : I, x

. A solution of (3.1) is a function u : 1, -f V which y) absolutely continuous, differentiab e a.e. on I,,

166 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS has its range in B (u ), and satisfies (3.1) a.e. on I,. Suf- bo ficient conditions for the Cauchy Problem for (3.1) to be well- posed are given in the following.

Theorem 3.1 Assume there is a pair of measurable functions 1 k(-) : 1, + (o,m) and Q(-) : Ia-f [l,m) with Q(*)/k(-) E L (Ia,R) such that

Then any two solutions u1(*), u2(-) of (3.1) will satisfy the estimate

In particular, the Cauchy problem of (3.1) with u(o) = uo has at most one solution.

Proof: Since k(t) > 0, the coercive estimate (3.2) implies

M(t) : V + VI is a bijection and 11 M(t)-’ IIL(,,lYv) S k(t1-l. Thus, we obtain from (3.1) llu;(t)-u;(t) /Iv 5 k(t1-l

11 f(t,ul(t))-f(t,u2(t)) 11 I 5 k(t)-lQ(t) ~~Ul(t)-u2(t)~~,,, a.e. V t E I,.

Since ul(*) u (0) is absolutely continuous with a - 2

167 SINGULAR AND DEGENERATE CAUCHY PROBLEMS summable derivative, we have Ilul(t)-u2(t)llV 5 tl ~/U~(O)-U~(O)~~~+ I I( ul(s)-u;(S)IIVds. Therefore the bounded 0 function Z(t) = Ilul(t)-u2(t)llv satisfies the inequality Z(t) 5

Z(o) + Itk(s)-’g(s)Z(s)dsy t E Ia.The estimate (3.4) follows 0 inmediately by the Gronwall inequality. QED

Theorem 3.2 In addition to the hypotheses of Theorem 3.1 , assume that t +. is measurable for each pair x, y E V and that t + is measurable for x E Bb(u0) and y E V.

Finally, let bo > 0 and c > 0 satisfy Ilf(t,uo)II 5 Q(t)boy a.e. rC V’ t E Icyand k(t)-’Q(t)dt 5 b/(bo+b). Then there exists a 0 (unique) solution u(*) of (3.1) on Icwhich satisfies u(o) = uo.

Proof: We first show that for every function u : Ic-+

Bb(uo) which is strongly (= weakly) measurable in V, the function t + Ll(t)-l 0 f(t,u(t)) : Ic-+ V is measurable. For each I$ E VI -1 * we have = . The indicated adjoint is measurable when M(t)-’ is, and so it suffices to show the measurability of each factor. To show f(t,u(t)) is measurable we consider first the case where u(*) is countably-

valued; thus u.(t) = x. for t E G where {G : j 2 1) is a J J j’ j measurable partition of I,. Letting @. denote the characteristic J function of G we obtain f(t,u(t)) = l{f(t,x.)I$.(t) : j f 1) jy JJ on I and this is clearly measurable. The case of general C’

u(*) follows from the strong continuity of x + P(t,x), since any measurable u is the strong lirrit of countably-valued rlieasurable

168 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS functions. To show the second factor is measurable, let m 2 1 and consider the restriction of M : 1, + L(V,V’) to the set Jm of those t €ICwith k(t) 2 l/m. Since M is measurable on this set,

there is a sequence of countably-valued functions Mk : Jm -+ M(J,) C L(V,V’) such that Lz Mk(t) = M(t) strongly for t E Jm. I V we have

on of 14-l to

desired re-

The proof of Theorem 3.2 now follows standard arguments.

Let X be the set of u E C(Ic,V) with range in Bb(uo). For u E X the function t * M(t)-lf(t,u(t)) is measurable Ic+ V (from above) and it satisfies the estimate IIM(t)-’ 0 f(t,u(t))lIv 2 k(t1-l -

Q(t) (bo+b) on I,. Hence, we can define F : X + X by [Fu](t) = t M(s)-lf(s,u(s))ds, t E I,, and it satisfies the estimate uo + 0 t (3.5) 11 [Ful(t)-[Fvl(t)llv 5 I k(s)-lQ(~)l~U(S)-V(S)I~~ds, 0

u,v E x, t E I,.

But any map F of a closed and bounded subset X of Lm(Ic,V) into itself which satisfies (3.5) is known to have a unique fixed- point, u (cf. Carroll [14]). This fixed-point is clearly the desired sol ution. OED

A solution exists on the entire interval 1, in certain

169 SINGULAR AND DEGENERATE CAUCHY PROBLEMS situations. Two such situations are given below.

Theorem 3.3 Assume all the hypotheses of Theorem 3.1 hold

with Bb(uo) = V. Also, suppose t + and t + are measurable Ia+ R for every pair x,y E V, and let uo E V, bo > 0 be given and satisfy Ilf(t,uo)llVI 5 Q(t)bo a.e. on I,.

Then there exists a unique solution of (3.1) on I, with u(o) = uo.

rt Proof: Set g(t) = bo exp{ k(s)-’Q(s)dsI and let X be those JO functions u E C(Ia,V) for which ]lu(t)-udl, 5 g(t) - bo for t E

I,. If u E X, the estimate Il~l(s)-lf(s,u(s))IIV 5 k(s)-’Q(s)(llu(s)-udlv + bo) shows that Fu defined as above belongs to C(Ia,V) and furthermore satisfies 11 [Fu](t)-udlv 5 t k(s)-lQ(s)g(s)ds = g(t)-bo, so Fu E X. Thus F has a unique 0 fixed-point u E X which is the solution of the Cauchy problem for (3.1). QED

Remark 3.4 The preceding results permit the leading operators to degenerate in a very weak sense at any to E I That is, a’ 1 (3.2) must be maintained with k(*)-’ E L (Ia,R) . On the other hand we have placed no upper bounds on the family {M(t) : t E Ial; they may be singular (cf. Introduction) on a suitable set of points in Ia.

The Lipschitz condition (3.3) together with the estimate on f(t,uo) in Theorem 3.3 impose a growth rate on f(t,u) which is linear in u. Such hypotheses are frequently appropriate, especially in linear problems such as Example 1.1 which occur

170 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS frequently in practice. However they are not appropriate for nonlinear wave propagation models, and these applications call for solutions global in time.

Example 3.5 The third order nonlinear equation

(3.6) Ut - Uxxt + ux + uux = 0 provides a model for the propagation of long waves of sma 1 ampli- tude. The interest here is in solutions of (3.6) for all t 2 0. Similarly, models with other forms of n.on1inearity in add tion to dispersion and dissipation occur in the form

u - u + a sgn(u) lulq + buPux = cu (3.7) t xxt xx'

We refer to Benjamin [l, 23, Dona [2,3, 4, 51, and Showalter [lo, 19, 211 for discussion of such models and related systems. Initial boundary value problems for (3.6) and (3.7) can be resolved in the following abstract forni.

Theorem 3.6 Let a > 0 and assume given for each t E 1, = I [O,a] an operator M(t) E L(V,V ), where V is a separable reflexive Banach space. Assume there is a k > 0 such that

each M(t) is symmetric, and for each pair x,y E V the function t + is absolutely continuous with

d 2 (3.9) dt 5 m(t)llxll , a.e. t E I,, x E V,

171 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

1 I for some m(-) E L (Ia,R). Let f: 1, x V -+ V be given such that

for each pair x,y E V the function t -+ is measurable, 1 and for each bounded set B in V there is a a(-) E L (IayR) such that Ilf(t,x)-f(t,y))I v~ 5 Q(t) Ilx-Alv, and Ilf(t,x)II 5 O(t), for x,y E B and a.e. t E I,. Finally, assume there exist K(-) 1 and L(*) E L (Ia)with L(t) 2 0 a.e. such that 5

K(t)/Ixlf + L(t)llXll, x E V. Then for each uo E V there exists a unique solution u : 1, + V of (3 1) with u(o) = uo.

Proof: Uniqueness and loca existence follow immediately from Theorem 3.1 and Theorem 3.2, respectively. The existence of a (global) solution on 1, can be obtained by standard continuation arguments if we can establish an a priori bound on a solution. But in the present situation the absolutely continuous I function a(t) 5 satisfies (cf. Lemma 5.1) a (t) =

Z + 5 [(ZK(t)+m(t))/k] a(t) +

C2L(t)/k’/2] o(t)”*; hence we obtain

’0 ’0

with H(t) = (K+(t) + (1/2)m(t))/k. The estimates (3.10) and (3.8) provide the desired bound on all solutions of (3.1) on 1,

with given initial data uo. OED

Remark 3.7 Equations similar in form to (3.G) have been

used to “regularize” higher order equations as a first step in

solving them (cf. Bona [4], Showalter [17]). The advantage over

172 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS standard (e.g. , parabolic) regularizations is that the order of the problem is not increased by the regularization. An important case of this is the "Yosida approximation" AE of a maximal accretive (linear) A in Banach space given by AE = (I+EA)-~A,E > I 0. Thereby, one approximates the equation u (t) + Au(t) = 0 by one with A replaced by the bbunded AEy or, equivalently, solves I the equation (I+EA)uE(t) + AuE(t) = 0 and then looks for 1 im u(t) = E4+ uE(t) in some sense. This technique was used for a nonlinear time-dependent equation by Kato [2] and to study the nonwell-posed backward Cauchy problem in Showalter [lG, 201; also see Brezis [3].

The preceding results immediately yield corresponding results for second order evolution equations of the form

I1 (3.11) M(t)u (t) = F(t,u(t),ul(t))

Theorem 3.8 Let the family of operators (M(t)> and the functions k(-) and Q(*) be given as in Theorem 3.2. Let uoyul E I V and the function F : 1, x Bb(uo) x Bb(ul) + V be given with

t + measurable for x1 E Bb(u0), x2 E Bb(ul). y E 2 2 V. Suppose we have IIF(t,x)-F(t,y)ll 5 Q(t)[llxl-YIIlv + IIx2-YdIv1~ VI 1) F(t,uoyul)l~v, 5 cl(t), and

for x = (xl,x2) andy = (y1,y2) in B (u ) x Bb(ul) and a.e. t E bo

Then there exists c > 0 and a unique u : Ic+ V which is Ia.

173 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

I continuously differentiable with u(t) E Bb(uo). u (t) E Bb(ul), I u is (strongly) absolutely continuous and differentiable a.e. , I (3.11) holds a.e. on Icand u(o) = uoy u (0) = ul.

The preceding follows from Theorem 3.2 applied to the equation M(t)U'(t) = f(t,U(t)) in V z V x V with M(t)x !

[M(t)xlyM(t)x2] and f(t,x) 5 [M(t)x2,F(t,xl ,x,)]. The estimate

(3.12) limits the growth rate of the leading operator (cf. Re- mark 3.4). This hypothesis can be deleted when V is a Hilbert

space; we need only replace M(t) by the Riesz map V -+ V' in the first factor of each of M(t) and f.

Initial boundary value problems for partial differential equations arise in the form (3.11) in various applications. Tww classical cases are given below; see Lighthill [l]or Whitham

[l]for additional examples of this type.

Example 3.9 The equation of S. Sobolev [l]

(3.13) A3utt + uzz = 0 describes the fluid motion in a rotating vessel. It is on ac- count of this equation that the term "Sobolev equation" is used in the Russian literature to refer to any equation with spatial derivatives on the highest order time derivative (cf., however,

Example 1 .2).

Example 3.10 The equation

(3.14) u aA3utt A3u = f(x,t) tt - -

174 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS was introduced by A.E.H. Love [l]to describe transverse vibrations in a beam. The second term in (3.14) represents radial inertia in the model.

Finally, we describe how certain doubly-nonlinear evolution equations

(3.15) M(t,u’(t)) = f(t,u(t)), 05t<_T, in Hilbert space can be solved directly by standard results on monotone nonlinear operators. The technique applies as well to the corresponding variational inequality

I I (3.16)

with a prescr bed fami y of closed convex subsets K(t) c V, so we consider this more general situation.

Let V be real Hilbert space and for each a 2 0 we let Va be the Hilbert space of square summable functions from [O,T] into T V with the norm Ilvlla = (1 Ilv(t)llie-2at dt)”2. For each t E 0 [O,T] we are given a pair of (possibly nonlinear) functionsf(t,*),

I M(t,*) from V into V . The following elementary result is fundamental for this technique.

Lemma 3.11 Assume [Mv](t) E M(t,v(t)) defines a hemicontin-

I uous M : Va + Va and that there is a k > 0 with

175 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

2 (3.17) 2 kllx-yllv, a.e. t E [O,T], x,y E V.

Let uo E V and assume [f(v)](t) = f(t,uo+ v(s)ds) defines f : I Va + V and that there is a Q > 0 with a

(3.18) 11 f(t,x) - f(t>Y)ll <- Q (1 X-Y 11 v> a.e. t E [O,T], x,y E V.

Then the operator T = M - f : Va -+ Va is hemicontinuous and strongly monotone for "a" sufficiently large.

Proof: For u,v E Va we have

The condition of uniform strong monotonicity on M(t,-) gives 2 2 c IIu-vII, , u,v E Va, hence we obtai'n 2

Ec-Q(T/~~)~/~I~~u-v~~~,u,v E V a' The desired result holds for a > TQ~/ZC~. QED

176 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Theorem 3.12 Let the hypotheses of Lemma 3.11 hold and assume in addition that we are given a family of nonempty closed convex subsets K(t) of V for which the corresponding projections

P(t) : V + K(t) are a measurable family in L(V) (cf. Carroll

[14]). Then there exists a unique absolutely continuous u E Vo = 2 I L (0,T;V) with u E Vo, u(o) = uo, and satisfying (3.16).

Proof: With a chosen as in Lemma 3.11 , the operator T : I Va * V is hemicontinuous and strongly monotone. Define K 5 a

{v E Va : v(t) E K(t), a.e. t E [O,T]). Then K is closed, convex and nonempty in Va, so it follows from Browder [5] (cf. Carroll

[14]) that there exists a unique w E K such that

(3.19) 2 0, v E K.

Finally, the measurable family of projections (P(t)> is used to show that (3.19) is equivalent to (3.16) with u(t) = uo i. rt Jow(s)ds, 0 5 t 5 T. QED

The preceding result is far from being best possible, but serves to illustrate the applicability of the theory of monotone nonlinear operators to the situation. A similar result holds for the doubly-nonl inear equation

and more genera ly, the neq ua ity

177 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

rt (3.21) 2 0, v E K(t), '0 - 1, u(t) E K(t), a.e. t E COYTI, under essentially the same hypotheses as Theorem 3.12. It would be desirable to have such results on either (3.16) or (3.21) with

M(t,-) permitted to be degenerate, or at least not uniformly bounded (cf. Remark 3.4). We shall return to similar problems in Banach space with a single linear operator (possibly degenerate) in Section 3.6 below. Also, a variation on Theorem 3.8

ng Theorem 3.12) gives existence results for second order

ution equations and inequalities.

Remark 3.13 Theorems 3.1, 3.2 and 3.3 are from Showalter

and Theorem 3.4 is from Showalter [18, 191. Theorem 3.8 was unpublished and Theorem 3.12 is due to Kluge and Bruckner [l].

For additional material on specific partial differential equations and on general evolution equations of the type considered in this section we cite the above references and Amos [l],Bardos-Brezis-

Brezis [Z], Bhatnager [l],Bochner-von Nuemann [l],Bona [l],

Brill [l],Brown [l],Calvert [l],Coleman-Duffin-Plizel [23,

Colton [l,2, 31, Davis [l,23, Derguzov [l],Dunninger-Levine

111, Eskin [l],Ewing [l,2, 31, W. H. Ford [l,21, Fox [31,

Gajewski-Zacharius [1, 2, 31, Galpern [l,2, 3, 4, 51, Grabmuller

[l],tiorgan-Wheeler [l],Ilin [l],Kostyuzenko-Eskin [l],Lagnese

[l,2, 3, 4, 5, 6, 7, 8, 91, Lebedev [l],Levine [3, 5, 6, 7, 8, 91, Lezhnev [l],Lions [5, 9, lo], Maslennikova [l,2, 3, 4, 5,

178 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

61, tkdeiros-Menzala [l],Mikhlin [Z], Fliranda [l],Neves [l],

Prokopenko [l],Rao [l,2, 4, 51, Rundell [l,2, 31, Selezneva

[l],Showalter [l, 7, 8, 9, 11, 12, 13, 141, Sigillito [l,2, 31,

Stecher-Rundell [l],Ting [l, 31, Ton [l],Wahlbin [l],Zalenyak

[l,2, 3, 41. One could also include certain references from Remarks 4.10, 5.18, 6.24 and 7.10 to which we refer for additional material.

3.4 Weakly regular equations We restrict our attention to a class of linear evolution equations of the form

(4.1) Mul(t) t Lu(t) = f(t), t > 0, and obtain we1 1-posedness results when M is not (necessarily) as strong as L. Although the semigroup techniques employed here will be used for nonlinear degenerate problems in Section 3.6 below, we introduce them in the simpler situation of (4.1) where we shall be able to distinguish the analytic situation from the

strong1y-continuous case.

Let Vm be a complex Hilbert space with scalar-product (-,*), and define the isomorphism (of Riesz) from V, onto its antidual I V, V, of conjugate-linear continuous functionals by = (x,y),,

x,y E V,. Suppose D is a subspace of Vm and the linear map I I

L : D -f Vm is given. u E V and f E C((o,m),Vm) are given, we If om 1 consider the problem of finding a u(*) E C([o,m), Vm) f) C ((o,m),

Vm) which satisfies (4.1) and u(o) = uo.

To obtain an elementary uniqueness result, let u(*) be a

179 SINGULAR AND DEGENERATE CAUCHY PROBEMS solution of (4.1) with f 0 and note that D,(u(t),u(t)),,, =

-2Re, t > 0. Hence, if L is monotone then Ilu(t)Ilm =

(u(t), u(t))A’2 is nonincreasing and a uniqueness result is obtained. Recall that L monotone means Re Z 0, x E D. This computation suggests that Vm is an appropriate space in which to seek well-posedness results. The equation in V lil -1 (4.2) ul(t) + M oLu(t) = M-lf(t), t>O is equivalent to (4.1) and suggests consideration of the operator -1 A = M OL with domain D(A) = D. We see from

(4.3) (Ax.Y), = , x E D, y E Vm that L is monotone if and only if A is accretive in the tlilbert space Vm. In this case, -A generates a contraction semigroup on Vm if I + A is surjective; then the Cauchy problem is well- posed for (4.2). These observations prove the following (cf.

Carroll [14], Hille [2], or Kato [l]).

I Theorem 4.1 Let M : Vn, * Vm be the Riesz map of the complex

I Hilbert space Vm, L : D -t V, be given monotone and linear with I domain in Vmy and assume M + L : D + Vm is surjective. Then for I each f E C([O,m),V,) and uo E D there is a unique solution u(-) of (4.1) with u(o) = uo.

The Cauchy problem for (4.1) is solved above by a strongly continuous semigroup of contractions. This semigroup is analytic when the operator A is sectorial (Kato [1]) and we describe this

180 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS situation in the following.

Theorem 4.2 Suppose M is the Riesz map of the Hilbert space

Vm, V is a Hilbert space dense and continuous in Vm, and L : V -+

V' is continuous, linear and coercive: Re L cllxIl;, x E V, I for some c > 0. Then for every Hb'lder continuous f: [O,..) -+ V, and uo E V, there is aunique, solution u(=) of (4.1) with u(o) = uO*

Each of the two preceding results implies well-posedness of the Cauchy problem for a linear second-order evolution equation

I1 I (4.4) Cu (t) + Bu (t) + Au(t) = f(t), t ' 0.

The idha is to write (4.4) as a first-order system in the form

Theorem 4.3 Let Va and Vc be complex Hilbert spaces with I

Va dense and continuous in Vc, and denote by A : Va -+ Va and C :

Vc -f Vc the respective Riesz maps. Let B be linear and monotone

I from D(B) C Va into Va and assume A + B + C : D(B) -+ Vi is sur- 1 jective. Then for each f E C (CO,~), VL) and pair uo E Va, u1 E I D(B) with Auo + Bul E Vc there is a unique solution u(*) E c([o,m),va) n c 1((oym),va) n cl([o,m),vc) n c2((o,m),vc) of (4.4)

I with u(o)) =uo and u (0) = ul.

I I I Proof: Define Vm = Va x Vc; then we have V, = Va x Vc and

181 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

I the Riesz map M : Vm + Vm is given by M[xl,x2] = [Axl,Cx2],

I [xl,x2] E Vm. Let D E {[x,,x,] E Va x D(B) : Axl + Bx2 E Vc), I I I recalling Vc C Va, and define L : D + Vm by L([xl ,x2]) =

[-Ax2,Axl + Bx2], [xl,x2] E I). With M and L so defined, the I system (4.5) is clearly equivalent to Mw (t) + Lw(t) = [O,f(t)], t > 0. In order to apply Theorem 4.1 we need only to verify that

L is monotone and M + L is surjective. But an easy computation shows Re = Re so B monotone implies I I L is monotone. Finally, if fl E Va and f2 E Vc, then we can solve (A+B+C)x2 = f2 - fly x2 E D(B), then set x1 = x2 + A-’fl to obtain [x, ,x2] E D with (M+L)[xl ,x2] = [fl ,f2]. QE D

Theorem 4.4 Let the Hilbert spaces and operators A: Va + I I I Va and C : Vc -+ Vc be given as in Theorem 4.3. Let B : Va + Va

I be continuous and linear with B + AC : Va + Va coercive for some I A > 0. Then for every H6lder continuous f : [O,m) + Vc and pair

E u1 E Vc, there is a unique solution u(=) E C([O,m),Va)n uo v a’ 1 1 2 c ((0,m),Va) fl c “O,m),Vc) n c ((O,m),Vc) of (4.4) with u(0) = I uo and u (0) = ul.

Proof: Following the proof of Theorem 4.3, we find that we can apply Theorem 4.2 with V = Va x Va if we verify that AM + L is V-elliptic for some X > 0. But AB + C being Va-elliptic is precisely what is needed. QED

Briefly we indicate some examples of partial differential equations for which corresponding initial-boundary value problems

182 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS are solved by the preceding results. The examples are far from best possible in any sense and are displayed here only to suggest the types of equations to which the results apply.

Example 4.5 Let m(*) E Lm(O,l) with m(x) > 0 for a.e. x E

(0,l) and define Vm to be the completion of C;(O,l) with the scalar-product (u,v) E m(x)u(x)mdx. Let V = HA(0,l) and m rl I 0 set = u.(x)vl(x)dx, u,v E V. Then Theorem 4.2 shows JO that the initial-boundary va 1ue problem

m(x)Dtu(x,t) - Dxu(x,t)z = f(x,t), t > 0,

u(0,t) = u(1,t) = 0,

U(X,O) = uo(x), O 0 in the space Vm of measurable functions v on (0,l) with 2 I2 m-1'2v E L (0,l). Note that Vm C L (0,l) CVm and the equation may be elliptic on a null set (where m(x) = 0).

1 -5--Exam le 4.6 Let Vm = Ho(O,l) with the scalar-product (u,~),,, = Jolu(x)W+ mul(x) vl(x))dx where m > 0. (More general 3 coefficients could be added as above.) Set Lu = Dxu on D E 2 1 {u E H (0,l)fl Ho(O,l) : c u'(0) = ~'(1))where c is given with

IcI 5 1. Then 2 Re = JuI (o)l - lu1(1)I2 2 0,u E D, so L is monotone; we can show M + L is surjective, so Theorem 4.1 shows the problem

183 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(Dt-mDxDt)u(x,t)2 + D,u(x,t)3 = f(x,t), 0 < x < 1,

u(0,t) = u(1,t) = 0, c ux(O,t) = ux(l,t), t 2 0,

u(x,o) = U,(X) is well-posed for appropriate f(*,*) and u This equation is a 0' 1 inear regularized Korteweg-deVries equation (cf. Bona [4]).

2 Example 4.7 Take Vm as above and let V = Ho(O,l) with L = 2 D: : V -+ V' defined by = 1; Dxu(x)Dxv(x)dx, u,v E V. Then Theorem 4.2 gives existence-uniqueness results for the metaparabolic (Brown [l])problem

(Dt mDxDt)u(x,t)2 + Dxu(x,t)4 = f(x,t), - O

= u(1,t) = ux(O,t) = ux(l,t) = 0, t > 0,

when uo E H:(0,1) and t + f(*,t) : [O,m) + H-'(O,l) is H6lder con tinuous.

1 rl Example 4.8 Let Va = Ho(O,l) with = dxy vc = 2 J," L (0,l) with = 11, uydx; positive coefficients could be added as in Example 4.5. Let r E $, b E R and define B: Va + Vc by Bu = ru + bul, u E Va. Since Re = Re(r) Ilully' , B is CI monotone whenever Re(r) 2 0. The operator A + B + C : Va + Va is continuous, linear and coercive, hence surjective, so Theorem 4.3 shows the problem

184 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

2 t)+rDtu( x,t)+bDxDtu( x,t)-Dxu( x,t) = f (x ,t) ,

(4.9) y0,t) = u(1,t) = 0, t> 0,

U(X,O) = u0(x), Dtu(x,o) = u 1(x), 0 < x < 1 ,

12 1 is well posed with uo E Ho"H (O,l), u1 E Ho(O,l) and t -+ 1 f(*,t) E C ([0,~),L~(0,1)). This is a classical problem for the damped wave equation. If instead of the above choice for B we set B = EA, E > 0, then from Theorem 4.4 follows well-posedness for the "parabolic" problem

rDtU(X,t)-EDXDtU(2 2 X,t)-Dxu(2 x,t) = f (x,t) , O

u(0,t) = u(1,t) = 0, t > 0,

u(x,o) = u0(x), Dtu(x,o) = u,(x), of classical viscoelasticity (cf. Albertoni-Cercignani [l]).Here 1 2 the data is chosen with u E Ho(OYl), u1 E L (O,l), and t -t 0 f(*,t) : [O,m) -+ L'(0,l) H6lder continuous.

We return to consider the parabolic situation of Theorem 4.2 and describe abstract regularity results on the solution. In addition to the hypotheses of Theorem 4.2, assume given a Hilbert space H which is identified with its dual and in which V, is dense and continuously embedded. Thus we have the inclusions VcVmC I I I 1 H = H C VmCV . Let M be the (unbounded) operator on H ob- I 1 tained as the restriction of M : Vm -t Vm to the domain D(M ) = I {u E V,,, : Mu E HI. Similarly, the restriction of L : V -t V to 1 the domain D (L ) = {u E V : Lu E HI will be denoted by L1. Each

1 85 SINGULAR AND DEGENERATE CAUCHY PROBLEMS of D(M1 ) and D(L 1 ) is a Hilbert space with the induced graph-norm and they are dense and continuous in Vm and V, respectively. We shall describe a sufficient condition for the solution u(*) of (4.1) with f(*) = 0 to belong to C([O,m),Vm)/)Cm((O,m),D(L 1 )).

The arguments depend on the theory of interpolation spaces and fractional powers of operators (cf. Carroll [14] for references).

Specifically, the operator L1 is regularly accretive and corresponding fractional powers Le, 0 5 0 L 1, can be defined. Their corresponding domains are related to interpolation spaces between 1 D(L ) and H by the identities D(Llme) = [D(L),H;O], 0 5 8 5 1, and corresponding norms are equivalent,

Theorem 4.9 In addition to the hypotheses of Theorem 4.2, 1 -e 1 we assume D(L ) C D(M ) for some 8, 0 < 8 5 1. Then for each uo E V, the solution u(*) of (4.1) with f(*)= 0 belongs to cm( (0,m) ,~d1)

Remark 4.10 Theorem 4.1 is new in the form given. The closely related Theorem 4.2 is from Showalter [6]; cf. Showalter

[7, 111 for an earlier special case. Theorem 4.3 is the linear case of a result in Showalter [5] and Theorem 4.4 is new.

Theorem 4.9 is presented in Showalter [6, pt. 111 with a large class of applications to initial-boundary value problems, The additional hypothesis in Theorem 4.9 makes L strictly stronger than M; the result is false in general when e = 0. For re ated work we refer to Coirier [l],Krein [l],Mikhlin [2], Phil ips

[l]and Showalter [17].

186 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

3.5 Linear degenerate equations. We shall consider appropriate Cauchy problems for the equation

whose coefficients {M(t)) and {L(t)) are bounded and measurable families of linear operators between Hilbert spaces. We show

(essentially) that the problem is well-posed when the {M(t)) are nonnegative, (L(t)+XM(t)) are coercive for some X > 0, and the operators depend smoothly on t. Some elementary applications to initial-boundary value problems are presented in order to indicate the large class of problems to which the results can be applied (cf. Examples 1.1, 1.4 and 1.6). Consideration of the case of (5.1) on a product space leads to a parabolic system which contains the second order equation (cf. Example 1.2) 2 (5.2) %C(t)u(t)) + $B(t)u(t)) + A(t)u(t) = f(t). dt

Certain higher order equations and systems can be handled similarly. In order to describe the abstract Cauchy problem, let V be I a separable complex Hilbert space with norm IlvlI; V is the antidual, and the antiduality is denoted by . W is a complex

Hilbert space containing V and the injection is assumed continuous with norm 5 1. L(V,W) denotes the space of continuous linear operators from V into W and T is the unit interval [0,1]. Assume that for each t E T we are given a continuous sesquilinear form

187 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

a(t;*,*) on V and that for each pair x,y E V the map t -+ R(t;x,y) is bounded and measurable. By uniform boundedness there is a number KR > 0 such that IR(t;x,y)l 2 KRllxll- llyll for all x,y E V and t E T. Standard measurability arguments then show that for 2 any pair u,v E L (T,V) the function t + a(t;u(t),v(t)) is integrable (cf. Carroll [14], p. 168). Similarly, we assume given for each t E T a continuous sesquilinear form m(t;*,*) on W such that for each pair x,y E W the map t +. m(t;x,y) is bounded and measurable. Let {V(t) : t E TI be a family of closed subspaces 2 of V and denote by L (T,V(t)) the Hilbert space consisting of 2 those (I E L (T,V) for which (I(t) E V(t) a.e. on T. Finally, let 2l uo E W and f E L (T,V ) be given. A solution of the Cauchy prob- 2 lem (determined by the preceding data) is a u E L (T,V(t)) such that

1 dt .+ m(O;uo,v(o) = lo 1 for all v E L2(T,V(t))/) H (T,W) with v(1) = 0.

A family {a(t;-,*) : t E TI of continuous sesquilinear forms on V is called regular if for each pair x,y E V the function t -t 1 a(t;x,y) is absolutely continuous and there is an M(-) E L (T) such that for x,y E V we have

a.e. t E T.

188 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

The following "chain rule" will be used below.

Lemma 5.1 Let {a(t;*,-) : t E T} be a regular family on V. 1 Then for each pair u,v E H (T,V) the function t -+ a(t;u(t),v(t))

is absolutely continuous and its derivative is given by

uta(t;u(t),v(t)) = al(t;u(t),v(t)) + a(t;ul(t),v(t)) + I a(t;u(t),v (t)), a.e. t E T.

Proof: Define a(t) E L(V,V) by (a(t)x,y),, = a(t;x,y), x,y E

V. Fix x E V and let{ yn : n Z 1) be dense in V. For each n 2 1 define (&(t)x,yn)v = Dt(a(t)x,yn)v, a.e:t E T. (The estimate

(5.4) shows (&(t)xyy)v is defined and continuous at every y E V and a.e. t E T.) The map &(*)x is weakly, hence strongly, mea-

surable and the estimate (5.4) shows it is in L'(T,V). The weak absolute continuity of t + a(t)x then shows a(t)x = a(o)x +

It&(s)x ds, t E T. Thus a(-)x is strongly absolutely continuous '0 and strongly differentiable a.e. on T. 1 Let u E H (T,V). For each v E V we have (a(t)u(t),v)v = * (u(t) ,a (t)vIv absolutely continuous, since the above discussion applies as well to the adjoint a"(t). Hence, t -+ a(t)u(t)

is weakly absolutely continuous. The strong differentiability of a(-)from above implies Dt[a(t)u(t)] = i(t)u(t) + a(t)ul(t), a.e. 1 t E T, hence the indicated strong derivative is in L (T,V). From this it follows that a(*)u(-) is strongly absolutely continuous and the desired result now follows easily. QED

189 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Our first result gives existence of a solution with an a priori estimate.

Theorem 5.2 Let the Hilbert spaces V(t) C V C W, sesquilinear forms m(t;*,-) and k(t;-,-) on W and V, respectively, 2' uo E W and f E L (T,V ) be given as above. Assume that

{m(t;-,=) : t E TI is a regular family of Hermitian forms on W: m(t;x,y) = 'myx,y E W, a.e. t E T, and m(o;x,x) 2 0 for x E W. Assume that for some real X and c > 0

(5.5) Rek(t;x,x) + hm(t;x,x) + m-(t;x,x)

2 2 CllXIl"' x E V(t), a.e. t E T.

Then there exists a solution u of the Cauchy problem, and it 2 1/2 satisfies IIuII 5 const.(IIfII I + m(o;uo,uo)) , L (T,V) L (T,V 1 where the constant depends on A and c.

Proof: Note first that by a standard change-of-variable argument we may replace k(t;-,-) by !L(t;-,*) + Xrn(t;*,=). There- fore we may assume without loss of generality that A = 0 in (5.5):

2Rek(t;x,x) + m'(t;x,x) 2 ~11x11~.x E V(t), ale. t E T. Now 2 1 define H = L2(T,V(t)) with the norm IIuIIH = Ilu(t)/12dt and let 0 2 F E {$ E H : $I E L2(T,W), $(1) = 0) with the norm 11011, =

190 3. DEGENERATE EQUATIONS WITH OEPRATOR COEFFICIENTS

sesquilinear, L : F -+ I$ is conjugate-linear, and we have the

continuous.

Finally, for 0 in F we have from Lemna 5.1 and 5.5 with A =

These estimates show that the projection theorem of Lions [5, p.

371 applies here. Thus, there is a u E H such that E(u,@) =

L(@) for all @ E F and it satisfies llullH 5 (2/niin(~.l))IIL~~~~.

But this is precisely the desired result. QED

Remark 5.3 The hypotheses of Theorem 5.2 not only permit m(t;x,x) to vanish but also to actually take on negative values.

This will be illustrated in the examples.

Remark 5.4 An easy estimate shows that (5.5) holds (with X replaced by 2X + a) if we assume that for some c > 0 and a 2 0

2 (5.7) ReR(t;x,x) + Xm(t;x,x) 2 cllxlIv,

This pair of conditions is thus stronger than (5.5) but occurs

frequently in applications. Note that (5.6) and m(o;x,x) 2 0

imply m(t;x,x) 2 0 for all t E T.

191 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

We present an elementary uniqueness result with a rather severe restriction of monotonici ty on the subspaces {V(t)>. This hypothesis is clearly "ad hoc" and is a limitation of the technique which is not indicative of the best possible results. In particular, much better uniqueness and regularity results for the very special case of m(t;x,x) = (x,x), have been obtained (cf.

Carroll [14], Carroll-Cooper [35], Carroll-State [36], Lions [5]); the novelty here is in the forms m(t;-,-) determining the leading

(possibly degenerate) operators (M(t)).

Theorem 5.5 Let the Hilbert spaces V(t) C V C W, sesquilinear forms !L(t;*,-) and m(t;*,*) on V and W, respectively, 2' uo E W and f E L (T,V ) be given as above. Assume that

Rem(t;x,x) 2 0, x E V(t), a.e. t E T, that {!L(t;=,*) : t E T) is a regular family of Hermitian forms on V(t):

x,y E V(t), a.e. t E T and for some real A and c > 0, R(t;x,x) + ARem(t;x,x) 2

~11x11~.x E V(t), a.e. t E T. Finally, assume that the family of subspaces {V(t) : t E TI is decreasing:

(5.9) t > T, t,T E T imply V(t)CV(-r).

Then there is at most one solution of the Cauchy problem.

Proof: Let u(-) be a solution of the Cauchy problem with

= 0 and f(*)= 0. By linearity it suffices to show that uO

192 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

u(-) = 0. Let s E (0,l) and define v(t) = - ['u(r)dr for t E 9 [o,s] and v(t) = 0 for t E [s,l]. Then v E L (T,V) and (5.9) t shows v(t) E V(t) for each t E T. Also, v (t) = -u(t) for t E

(0,s) and v'(t) = 0 for s E (O,l), so we have v' E L2(TyV(t))C 2 L (T,W) and v(1) = 0. Since u(.) is a solution we have by (5.3) ,flR(t;v'(t),v(t))dt - 1: rn(t;u(t),u(t))dt = 0. Lemma 5.1 and (5.8) give us the identities j12 Rem ( t ;u ( t ) ,u ( t ) ) d t = 1: Dt%( t ;v ( t ) ,v ( t ) ) - R ' ( t ;v ( t ) ,v ( t ) ) 1d t , rs I {ZRem(t;u(t),u(t)) + R'(t;v(t),v(t))}dt + Q(o;v(o),v(o)) = 0. '0

As before, we may assume %(t;x,x) L ~11~~~~.x E V(t), t E T. Define W(t) = 1: u(.r)d-r; then W(s) = -v(o) and W(t) - W(s) = v(t) for t E (0,s) and we obtain the estimate cllW(s) 11' 5 %(o;W(s),W(s)) + j: ZRem(t;u(t),u(t))dt

S 5 2 ,f M(t){llw(t)l12 + I/W(s)/121dt. 0 S Choose s > 0 so that 2 M(t)dt < c. Then for s 8 [o,so] we 0 1 0 rS have IIW(s) 11' 5 (2/(c - 2 OM(t)dt)> jS14(t)llW(t) 1I2dt. From JO 0 the Gronwall inequality we conclude that W(s) = 0 for s E [oyso]. hence u(s) = 0 for s E [o.so]. Since M(*) is integrable on T we could use the absolute continuity of the integral

193 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

T+ So choose so > 0 in the above so that I M(t)dt < c/2 for every T T 2 0 with T + so 5 1. Then we apply the above argument a finite number of times to obtain u(-) = 0 on T. QED

A fundamental problem with degenerate evolution equations is to determine the precise sense in which the initial condition is attained. A consideration of the two cases of m(t,x,y) : (X,Y)~ and m(t;x,y) = 0 shows (see below) that we may have the case of u(t) = u in an appropriate space or, respectively, that Lzt 0 lim+ u(t) does not necessarily exist in any sense. These remarks t-4 illustrate the importance of the following.

Theorem 5.6 Let u(*) be a solution of the Cauchy problem and assume the subspaces {V(t) : t E TI are initially decreasing: there is a to E (0,1] such that (5.9) holds for t,T E [o,to].

Then for every T E (o,to) and v E V(T) we have t + m(t;u(t),v) 1 is in H (0,~)(hence is absolutely continuous with distribution 2 derivative in L (0,~))and m(o;u(o) - uo,v) = 0. Thus, if

(){V(T) : T > 0) is dense in W, then m(o;u(o) - uo,u(o) - uo) = 0.

Proof: Define v(t) = for t E (o,T) and v(t) = 0 for 2 t E CT,~]. Then v(*) E L (T,V(t))f) H1(T,W) and v(1) = 0, so

That is, we have the identity

(5.10) R(*;u(-),v) t Dtm(*;u(*),v) =

194 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS in the space of distributions on (0,~)for every v E V(T). The 2 first and last terms of (5.10) are in L (0,~)and so, then, is the second and we thus have pointwise values a.e. in (5.10) and hence the equation

= jlg( t)dt

2 for g E L (0,~)and v E V(r). 1 Suppose now that $ E H (0,~)is given with $(T) = 0. Define v(*) as above so as to obtain from (5.3) 1: R(t;u(t),v)$(t)dt - /)(t;u(t),v)$'(t)dt = 1: $(t)dt t m(o;uo,v)$(o) where v E V(T). Since (5.11) holds with g(t) = $(t), we obtain from these r{Dtm(t;u(t) ,v)$(t) t m(t;u(t),v)$'(t)Idt = -m(o;uo,v)$(o). 0 The integrand is the derivative of the function t -+ 1 m(t;u(t),v)$(t) in H (0,~)and $(r) = 0, so the desired result follows. QED

Another situation which arises frequently in applications and to which the above technique is applicable is the following.

Theorem 5.7 Let u(*) be a solution of the Cauchy problem and assume there is a closed subspace Vo in V with VoCf){V(t) : t E TI. Define the two families of linear operators {L(t)) C

L(V,Vi) and {M(t)ICL(W,Vi) by = R(t;x,y), x E V, y E

Vo, t E T; = m(t;x,y), x E W, y E Vo, t E T. Then in

195 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

I the space of Vo-valued distributions on T we have (5.1) and the I function t -f M(t)u(t) : T -t Vo is continuous.

The proof of Theorem 5.7 follows from (5.10) with v E Vo and the remark that each term in (5.1), as well as l.I(-)u(-), is in

Our last result concerns variational boundary conditions.

Suppose we have the situation of Theorem 5.7 and also that

(5.9) holds. Let H be a Hilbert space in which V is continuously I imbedded and Vo is dense. We identify H with H by the Riesz I theorem and hence obtain Vo+ H4Vo and the identity = 2 (h,v)H for h E H, v E Vo. Assume f E L (T,H), T > 0 and v E V(T).

Then from (5.11) and (5.1) we obtair: R(t;u(t),v)@(t)dt + rT 1: Dtm(t;u(t) ,v)$(t)dt = (L(t)u(t) + Dt(M(t)u(t)) ,V)H$(t)dt for JO each @ E C;(O,T). This gives us the following.

Theorem 5.8 Assume the situation of Theorem 5,7 and that 2 (5.9) holds. Let H be given as above and f E L (T,H). Then for 2 each T > 0 and v E V(T) we have in L (0,~)

(5.12) R(t;u(t),v) + Dtm(t;u(t),v)

We briefly indicate how the preceding theorems can give corresponding results for second order evolution equations which are

"parabol ic". These results are time dependent analogues of Theo- rem 4.4.

196 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Theorem 5.9 Let Va and Vc be separable complex Hilbert spaces with Va dense and continuously imbedded in Vc. Let

{a(t;=,-) : t E TI, {b(t;-,-) : t E TI and {c(t;*,-) : t E TI

be families of continuous sesquilinear forms on Va, Vay and Vcy

respectively. Define corresponding fami J ies of 1inear operators {A(t)l and {B(t)IcL(Va,V:) and {C(t)> CL(Vc.Vi) as in Theorem

5.7. Assume that {a(t;*,-) : t E TI and (c(t;*,*) : t E TI are regular families of Hermitian forms with a(o;x,x) 2 0 and c(o;x,x) t 0 for x E Va. Assume that for some X and c > 0 I 2 Xa(t;x,x) + a (t;x,x) 2 cllxllvay

I 2 (5.14) 2Reb(t;x,x) + Xc(t;x,x) + c (t;x,x) t cIIxII , 'a for each x E Va and a.e. t E T. Then for uo E Va, vo E Vcy and 2l f(-),g(.) E L (T,Va) given, there exists a pair u(*),v(*) E

L2 (T,Va) which satisfies the system

I in the sense of Va-valued distributions on T and satisfies the initial conditions A(o)u(o) = A(o)uo, C(o)v(o) = C(o)vo.

Proof: This follows directly from Theorem 5.2 and Theorem

5.7 with m(t;X,?) f a(t;xl ,yl) + c(t;x2,y2) for X,y E W E Va x Vc and !L(t;Y,a = a(t;x,,y2) - a(t;x2,yl) + b(t;x2,y2) for ~,TE v V(t) v0 5 va x va. QED

197 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

There are two natural reductions of the first order system

(5.15) to a second order equation, and we shall describe these.

First, we set g( 0) = 0 in Theorem, 5.9 and note than that both 1' C(-)v(*) and Dt(C(*)v(*)) + B(*)v(.) belong to H (T,Va). Thus v(=) is a solution of the second order equation

and the initial conditions (Cv)(o) = C(o)vo, (Dt(Cv) + Bv)(o) =

-A(o)uo. Second, we set f(-) = 0 in Theorem 5.9 and assume A(t) is independent of t, i.e., A(t) 5 A for some necessarily coercive

I and Hermitian A E L(Va,Va). From (5.15) it follows that u(-) E 1 H (T,Va) and is a solution of the second order equation

(5.17) Dt(C(t)u'(t)) + B(t)ul(t) t Au(t) = g(t)

I and the initial conditions u(o) = uo, (C(*)u )(o) = C(o)vo.

We shall present some elementary applications of the preceding results to boundary value problems for partial differential equations in the form (5.2). These examples will illustrate some limitations on the types of problems to which the theorems apply.

Example 5.10 For our first example we choose Vo = V(t) = 1 1 V = Ho(T), W = {@E H (T) : @(1) = 03 and H = L2(T). Recall the estimate

198 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Let a : T +. T be absolutely continuous and define the Hermitian forms m(t;@,$) = ~~(t)@(x)~dxy@,$ E W, t E T. Then m(*;@,$) I is absolutely continuous and satisfies m (t;@,$)= '1 I@(a(t))12a'(t), @ E W, a.e. t E T. Since a E L (T), the estimate (5.18) shows that this is a regular family of forms. 11 For @,$ E V, we set !L(t;@,$) = 1 @ (x)$'(x)dx, t E T, From the 0 we1 1 -known inequality

it follows that the preceding family satisfies (5.7) with A = 0.

Finally, let uo E W and F E L2(T x T) be given and set f(t) =

F(-,t), t E T. Then the hypotheses of Theorem 5.5 (on uniqueness) are satisfied. Suppose additionally there is a number a,

0 < a < 2, such that

I (5.20) (t) 2 a - 2, a.e. t E T.

I I2 Then (5.18) shows that 2!L(t;@,@) + m (t;@,@)2 (2-0)11@ 1) 2 , L (TI @ E V, so by (5.19) it follows that (5.5) holds with A = 0, hence, the hypotheses of Theorem 5.2 (on existence) are satisfied.

Let u(*,t) = u(t) be the solution of the Cauchy problem.

Then u(x,t) is the unique generalized solution of the elliptic- parabolic boundary value problem, ut - uxx = F(x,t), 0 < x < a(t); -uXx = F(x,t), a(t) < x < 1; u(O,t) = u(l,t), t E T; u(x,o) = uo(x), 0 < x < a(o). The solution u(t) belongs to ti2(T) for each t E T, so u(*,t) and ux(-,t) are continuous

199 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

across the curve x = a(t), t E T.

Remark 5.11 The estimate (5.20) permits a(-) to decrease, but not very rapidly, whereas the estimate (5.6) holds if and I only if a (t) t 0 a.e. on T. The curve x = a(t) is noncharac- teristic a.e. on T; the example a(t) = 1/2 + [(t - 1/2)/4] 1/3 shows it may actually be characteristic at certain points.

Remark 5.12 The conclusions of Lemma 5.1 hold ifwe assume only that a(*;x,y) is absolutely continuous for each pair x,y E

V. Hence we need not assume an estimate like (5.4) on the family

{m(t;.,-)> in order to obtain Theorem 5.2. One can prove this assertion as follows: (1) use the closed graph and uniform boundedness theorems to obtain an estimate

'0 I 2 (2) approximate u in L (T,V) by simple functions and use the

Lebesgue theorem with (5.21) to obtain the result for the special

I case of constant v E V; approximate v by step functions and use the results of (1) and (2) to obtain the general result. The details are standard but involve some lengthy computations.

Example 5.13 Our second example is similar but allows the equation to be of Sobolev type in portions of T x T. Choose the spaces and the forms a(t;-,*) as before. For Q,$ E W, define r@(t) I I m(t;Qy+3) =r(t)Q(x)mdx + Jo 4 (x)+ (x)dx, t E T, where ci 0

200 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS and B map T into itself, both are absolutely continuous, and B is nondecreasing. For $I,$ E W, m(-;$I,$) is absolutely continuous so Remark 5.12 applies here. Using the estimate sup{l$I(s)12 : 0 5 s 5 B(t)) 2 ~~(t)l$I'(x)12dx,$I E V, one can I show that the estimate (5.5) holds if a (0) has an essential lower bound (possibly negative) and if there is a numberu, 0 < u < 2, I such that a (t) 2 u - 2 where a(t) > B(t). Thus, for each uo E

W and F E L2(T x T) it follows from Theorems 5.2 and 5.5 that there exists a unique generalized solution of the elliptic- parabolic-Sobolev boundary value problem, ut - uxxt - uxx = F,

0 < x < a(t), x < B(t); -Uxxt - uxx = F, a(t) < x < B(t); - uXx = F, B(t) < x < a(t); -uXx = F, a(t) < X, B(t) < x < 1; u(0,t) = u(1,t) = 0, t E T; u(x,O) = uo(x). 0 < x < max{a(o),

B(o)). Examples (e.g., a(t) E 0) show that if B is permitted to decrease, then a solution exists only if the initial data uo satisfies a compatibility condition. This is to be expected since the lines "x=constant" are characteristic for the third order Sobolev equation.

Example 5.14 We shall seek conditions on the function m(x,t) which are sufficient to apply our abstract results above to the boundary value problem a/at(m(x,t)u(x,t)) - uxx = F(x,t),

(x,t) E T x T; u(O,t) = u(1,t) = 0; ~(x,o)(u(x,o)-u~(x)) = 0, 1 a.e. x E T. We choose the spaces Vo = V(t) = V = Ho(T) and the 2 forms k(t;=,-) as above and let H = W = L (T). Assume the real valued m(-,*) E Lm(T x T) is such that m(x,-) is absolutely

201 SINGULAR AND DEGENERATE CAUCHY PROBLEMS continuous for a.e. x E T and satisfies Imt(x,t)I 5 M(t) for a.e. 1 t E T, where M(-) E L (T). Then it follows that the family of

Hermitian forms m(t;$,$) : m(x,t)$(x)mdx, $,$ E W, is re- 2 1: gular. Let uo E H, F E L (T x T), and set f(t) = F(*,t).

Theorems 5.6 and 5.7 show that the boundary value problem above is the realization of our abstract Cauchy problem.

In order to obtain uniqueness of a generalized solution from

Theorem 5.5, note that the forms !L(t;*,*) are coercive over V, so Theorem 5.5 is applicable if and only if additionally we as- s ume

(5.22) m(x,t) L 0 x,t E T, since this is equivalent to m(t;=,-) 2 0 for all $ E W, t E T.

To obtain existence from Theorem 5.2 it suffices to assume

m(x,o) 2 0, a.e. x E T, and

(5.23) 2 ess inf{mt(x,t) : (x,t) E T x TI > -2~r , for then we obtain the estimate m'(t;4,$) 2 (a - 2~r')Il2 I$l , 2 0 41 E V, for some number 0, 0 < a < ZIT . Then from (5.19) we ob- 2 tain (5.5) with X = 0 and c = a/2~. Thus, (5.22) gives uniqueness and (5.23) implies existence of a generalized solution of the problem.

Clearly neither of the conditions (5.22), (5.23) implies the other. In particular (5.23) permits m(x,t) to be negative;

202 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS then the equation is backward parabolic. This is the case for the equation

(5.24) ~(~‘/4)(1-2t)~u(x,t))= uxx for which mt(x,t) = -(3~2 /2)(1-2t)‘; hence, (5.23) is satisfied and existence follows. But m(=,.) is nonnegative only when

0 i t 1/2, so uniqueness is claimed only on this smaller interval. To see that uniqueness does not hold beyond t = 1/2, note that u(x,t) = (~2t)-~e~p{l-(1-2t)-~} sin (TX) is a solution of (5.24); a second solution is obtained by changing the values of u(x,t) to zero for all t > 1/2.

The leading coefficient in the equation

(5.25) $$(T 2 /N)(1-t2)’/2u(x,t)) = uxx (N > 0)

2 satisfies (5.22), hence uniqueness follows, but mt(x,t) > -27~ only on the interval [0, 1-(4N)-‘). Hence, the existence of a solution follows on the interval [O, 1-(4N)-2-u] for any u > 0; choosing N large makes the interval of existence as close as -1/2 desired to [O,l]. However the function u(x,t) = (l-t) exp(2N(1-t)’/2) sin (TX) is the unique solution of the problem with uo(x) = sin (TX) on any interval [0, 1-u], u > 0, and this 2 function does not belong to L (T x T). Hence, there is no solution on the entire interval.

Example 5.15 Let G be a smooth bounded domain in Rn and r(x) be a smooth positive function on G such that for those x

20 3 SINGULAR AND DEGENERATE CAUCHY PROBLEMS near the boundary aG, r(x) equals the distance from x to aG. Parabolic equations of the form

(5.26) tut + r(x)L(t)u = F(x,t), XEG, tET arise in mathematical genetics (cf. Kimura and Ohta [l], Levikson and Schuss [l]). They are typically degenerate in t (at t = 0) and degenerate in x (at x E 3G); L(t) is a strongly uniformly elliptic operator. We indicate how such problems can be included in the preceding theorems. Suppose we are given real- valued absolutely continuous R. .(t)(l 0, z E qn. Take Vo = V = Ho(G) and define a regular family n of coercive forms by R(t;@,$) = { 1 Rij(t)Di@D.$ + n G i ,j=1 J 1 R.(t)Dj$ $)dx, @,I) E V. The corresponding elliptic operators j=o J are given by n n (5.27) L(t) = - 1 Rij(t)DjDi + 1 R.(t)D t E T. i ,j=l j=o J j'

Let m(x,t) be given as in Example 5.14, set W = Ho(G),1 and define m(t;@,$) = 1 (m(x,t)/r(x)2)@(x)mdx,$,$ E W. From the inequal- G i ty

(5.28) II@/rll 2 5 Cll@llW, $ E W , L it follows that m(t;*,*) is a regular family of Hermitian forms on W. Let F E L2(G x T) and define f(t) = F(m,t)/r(*) for t E T. 2' From (5.28) it follows that f E L (T,V ). Finally, assume uo E

204 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Hi(G). From Theorems 5.6 and 5.7 we see that our abstract Cauchy problem is equivalent to the initial-boundary value prob-

1 em

L(wl(x,t))at r x + r(x)L(t)u(x,t) = F(x,t), x E G,t E T

s E aG,

m(x,o)(u(x,o)-uo(x)) = 0, x E G.

(The first equation has been multiplied by r(x) and the last by r(x)' > 0.) Existence and uniqueness follow from the additional assumptions m(x,t) 2 0, mt(x,t) 2 0, a.e. x E G, t E T. These are not best possible for existence; they can be weakened as in

Example 5.14.

Remark 5.16 The equation (5.29) permits the leading operat- or to be singular and the second to be degenerate in the spatial variable, while the leading operator may be degenerate in time.

Remark 5.17 A minor technical difficulty arises from the form in which (5.26) is given with the time derivative on u, not on (tu). We first note that tut = (tu), - u but see that the resulting second operator L(t) - u/r might not be coercive.

However, in the change of variable T = t" the leading term is given by tut = C~TU = "(TU), - au and the second operator result- T ing as above is given by L(T~/") - au/r. From (5.28) it follows that this last operator is coercive for CI > 0 sufficiently small.

205 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Remark 5.18 Theorems 5.2, 5.5, 5.6, 5.7 and 5.8 are from

Showalter [2]. Theorem 5.9 which contains (5.16) and (5.17) is new. See Showalter [Z] for examples in higher dimension with elliptic-parabolic interface or boundary conditions and other constraints on the time-differential along a submanifold or por- tion of the boundary. For additional material on specific partial differential equations or evolution equations of the type considered in this section we refer to Baiocchi [l],Browder [4],

Cannon-Hi 11 [Z], Fichera [2] , Ford [l] , Ford-Wai d [2] , Friedman- Schuss [l],Gagneux [l],Glusko-Krein [l],Kohn-Nirenberg [l],

Lions [5, 91, Oleinik [2], Schuss [l],Vicik [110 Also see references given in preceding sections for related linear problems and in the following sections for nonlinear problems.

3.6 Semilinear degenerate equations. We shall present some methods for obtaining existence-uniqueness results for the abstract Cauchy problem for the equation

where M is continuous, self-adjoint and monotone. The mst we shall assume on the nonlinear N is that it is monotone, hemicontinuous, bounded and coercive. Specifically; 1et.V be a reflex- I ive real Banach space with dual V and let N be a function from

I V to V . Then N is said to be monotone if

(6.2) 2 0, u,v E v,

206 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

hemicontinuous if for each triple u,v,w E V the function s -+

is continuous from R to R, bounded if the image of each bounded set in V is a bounded set in V-, and coercive if for each v E V

(6.3) lim (/llull) = +a0 . II u ll- Let H be a Hilbert space in which V is dense and continuously imbedded. Identify H and its dual and thereby obtain the inclu- I sions V C H C V . We suppose M is a continuous, self-adjoint and monotone operator of H into H; thus the square root M112 of M is defined. Let p ,> 2, l/p + l/q = 1, 0 < T < 00, and define V = 2 '2 LP(o,T;V), H = L (o,T;H), V = L (o,T;VI). Let Lou = dMu/dt for I u E D(Lo) = {u E V : du/dt E HI and denote by L : D(L) E V the closure of L in V x V'. A standard computation shows that if 0 u E D(L) then M1/2u E C(o,T;H), the space of continuous H-valued functions on [o,T].

Lemma 6.1 For u E V and f E V , the following are equivalent:

(a) u E D(L), M1/2u(o) = 0, and Lu = f,

for all v E V with dv/dt E H and v(T) = 0.

Proof: Clearly (a) implies (b): we need only observe that the identity in (b) holds for u E D(Lo), than pass to the

207 SINGULAR AND DEGENERATE CAUCHY PROBLEMS general case of u E D(L) by continuity. Conversely, if (b) is rt true then we define un(t) = n e n(s-t)u( s)ds, fn( t) = t JO nj en(s-t)f(s)ds. Then un E D(L0), Loun = fn, and we have 0 I (un,fn) -+ (u,f) in v x v . QED

The basic idea here is to consider the linear operator A :

D(A) -+ V’ defined by AU = Lu and D(A) = {u E D(L) : M1I2u(o) = 01.

That is, A = (d/dt)M with zero initial condition. Clearly A is closed, densely-defined and linear. Lemna 6.1 shows A is the adjoint of the operator -Md/dt with domain {v E V : dv/dt E H, v(T) = 0). That is, A is the adjoint of a monotone operator, so * A , be ng the closure of a monotone operator, is necessarily monotone. Since the above implies that A is maximal monotone by a theore of Brezis [I]we obtain the following from Browder [l]: I for N : V -+ V monotone, hemicontinuous, bounded and coercive, the operator A + N is surjective. The preceding remarks give the following result.

I Theorem 6.2 Let the spaces V c ti C V , the self-adjoint and monotone M E L(H) and the monotone, hemicontinuous, bounded and coercive 14 : V -+ V1 be given. For each f E Lq(0,T;V1) there exists a u E LP(o,T;V) such that

rT T rT (6.4) + / = Jo -1 0 0 2 for all v E Lp(oYT;V) with dv/dt E L (o,T;ti) and v(T) = 0. Also

M112u E C(o,T;H) and M1l2u(o) = 0. The solution is unique if N

208 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

is strictly monotone, i.e., if > 0, x,y E V, x + y.

Remark 6.3 The condition expressed by (6.4) is equivalent

to the Cauchy problem

$u(t) + N(u(t)) = f(t) (6.5) (M’j2u) (0) = 0

in which the equations hold in Lq(0,T;V’) and in H, respectively.

Remark 6.4 The extension of Theorem 6.2 to the case of

initial data u E H (not necessarily homogeneous) is given in 0 Brezis [l]. Then a term (M’/2~o,M1/2~(~))His added to (6.4) 1/2 and the initial condition in (6.5) becomes M’12u(o) = M uo. Also, the initial condition in Theorem 6.2 can be replaced by

the periodic condition M1/2u(o) = M1/’u(T) by an obvious modi-

fication of the operator A.

We consider briefly the Cauchy problem that arises when the

solution is constrained to a specified convex set. This leads

to a variational inequality. For such problems, the “natural”

I hypotheses on the nonlinear operator N : V + V is that it be

pseudomonotone: if lim u = u in V weakly and lim sup 5 0, then for every v E V j

5 lim inf . (6.6) JJ

Remark 6.5 The pseudomonotone operators include the semi-

monotone operators of Browder [6], the operators of calculus-

209 SINGULAR AND DEGENERATE CAUCHY PROBLEMS of-variations introduced by Leray and Lions [l],and the demi- monotone operators of Brezi s [4]. The pseudomonotone operators are precisely those which are most natural for elliptic variational inequalities.

To indicate the relationship between Theorem 6.2 and results to follow, we shall prove that every hemicontinuous and monotone I operator N : V + V is pseudomonotone. So, suppose the net Iu.1 J in V satisfies lim u. = u (weakly) and lim sup 5 0. J JJ Since N is monotone it follows that 2 . JJ J The right side converges to zero so we have lim = 0. JJ Let v E V. For 0 < t < 1 and w = (1-t)u + tv we have by monotonicity 2 0 and this shows t + J J J + t 2 . Taking the lim inf J and dividing by t gives lim inf 2 . The JJ result follows by using hemicontinuity to let t + 0 above.

A variational inequality problem corresponding to (6.5) and a given convex subset K of V is the following: find u E K with

M1/*u(o) = 0 for which

(6.7) MU) + N(u) - f,v-u>dt 2 0 0 for all v E K. A technical difficulty with this formulation is that the functional equation (6.7) might not imply that (d/dt)Mu I E V , so we shall weaken the problem appropriately. If u is a I solution of (6.7) and if v E K with (d/dt)Mv E V then we obtain

21 0 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

ITd ITd dt = <-( MU)+Nu-f ,v-U>dt + dt :t Jo 1, dt I’0 2 (l/E)l;. The last term is nonnegative if we require Mv(o) = Mu(o) = 0. Under hypotheses to be given below, we shall consider the (weak) variational inequality

T (6.8) u E K, I <&(Mv)+Nu-f,v-u>dt 2 0 0 I for all v E K such that (d/dt)Mv E V and Mv(o) = 0. I I Let the spaces V C H C V and V C ff C V be given as above.

Let K(t) be a closed convex subset of V for each t E [o,T]; assume 0 E K(t) and K(s) c K(t) for 0 <, s <- t <, T. Define K =

{v E V : v(t) E K(t) a.e.l.

The semigroup of right shifts given by

(6.10) G(s)KC K by the assumption that the convex sets are increasing. Denote by I -L the generator of (G(s) : s 2 01 on V(H,V ) and its domain by

D(L,V) (respectively, D(L,ff), D(L,V’)). Thus, we have L = d/dt with null initial condition in each of V, H, and VI. From (6.10) and the representation for the resolvent of -L by the formula cw (I+eL)-’ = E-’exp(-t/E) G(t)dt we obtain 0

21 1 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(6.11) (I t EL)-~KcK, E > 0,

I Suppose M E L(V,V ) is nonnegative and self-adjoint and define I M E L(V,V ) by [Mv](t) = Mv(t). Then we consider the operator I defined by A = L o M, D(A) = {v E v : Mv E D(L,V )>. The plan is to apply Corollary 39 of Brezis [4] to solve the abstract variational inequality

(6.12) u E K, 2 0 for all v E D(A) n K. The problems (6.8) and (6.12) are certainly equivalent with our choice of data and we are led to the following result.

Theorem 6.6 Let the spaces V c H C V' and the self-adjoint I and monotone M E L(V,V ) be given as above. Let K be the closed convex set in V obtained as above from an increasing family

{K(t) : 0 S t 5 TI of closed convex subsets of V, and assume 0 E I K. Let N : V -+ V be pseudomonotone, bounded and coercive. Then for each f E Lq(o,T;V') there exists a solution of (6.8). The solution is unique if N is strictly monotone.

Proof: The result follows imnediately from Brezis [4] after we verify that the linear A is nonnegative (= monotone) and com- I patible with K. Note first that for those v E V with Mv E V we have j:$Vv,v>dt = (l/B)(~.T Thus for v E D(A) it fol1 ows

21 2 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

(6.13) jIdt = (1/2) >_ 0.

That A is compatible with K means that for each u E K there exists Cunl C Kfl D(A) such that lim(un) = u in V and lim sup 5 0. Thus, let u E K and define for n 2 1 -1 -1 u n = (I+n L) u. We have lim (un ) = L(, u n (t) = n and (6.11) shows un E K. The representation above gives t Mun(t) = nj en(s-t)Mu(s)ds, hence, LMu, = MLun. Finally, since 0 -1 M is monotone we have = - =

-n-l 5 0, so A is compatible-with K. QE D

Remark 6.7 If we change the domain of L appropriately, we can replace the initial condition implicit in (6.8) by a corresponding periodic condition.

We return now to the Cauchy problem for the evolution equation (6.1) where M is given as in Theorem 6.6 but with weaker restrictions on the nonlinear operator N. Specifically, we shall

I assume N : V + V is of type M: if lim (u.) = u in V weakly, J I lim (Nu.) = f in V weakly, and lim sup <_ , then J jj Nu = f.

Remark 6.8 The operators of type M include weakly continuous operators (i.e., continuous from V weakly into V’ weakly) and the pseudomonotone operators, hence, the operators of Remark

6.5.

We show that pseudomonotone operators are of type M. Let

21 3 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

{u.) be a net (or sequence if V is separable) as in the definition J above of "type M" and assume N is pseudomonotone. Then lim sup 5 0, so for every v E V we obtain from (6.6)

5 lim inf 5 . But v E V is arbitrary JJ so Nu = f follows from this inequality. Much of the notation of Theorem 6.6 will be used to prove the following result.

I Theorem 6.9 Let the spaces V C H C V and the monotone I self-adjoint M E L(V,V ) be given as in Theorem 6.6. Let N :

V -f V be type M, bounded and coercive. Then for each f E

Lq(o,T;VI) there exists a solution u E LP(o,T;V) of the Cauchy problem

dMU) + N(u) = f (6.14)

(Mu)(o) = 0.

If N is strictly monotone (cf. Theorem 6.2) the solution of

(6.14) is unique.

Proof: We give a finite-difference approximation to (6.14) of implicit type. Thus, (6.14) is approximated by a corresponding family of "stationary" problems (cf. Chapter 11.7 of Lions

[lo]). Note that the Cauchy problem (6.14) is equivalent to the I equation in V

(6.15) A(u) + N(u) = f, A=LoAl

21 4 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS where -L is the generator of the strongly-continuous semigroup

I I (6.9) in V and M E L(V,V ) is obtained (pointwise) from M as before.

To approximate (6.15) we let s > 0 and consider the equa-

l tion in V

(6.16) [(I-G(s))/s]Mu, + NuS = f.

Since m(x,y) = is a semi-scalar-product on V we obtain the estimate 1: dt = 1: m(v(t-s) ,v(t)dt 5 1: dt, v E V, which shows that the operator [I-G(s)]M is monotone (cf.

6.13). Thus, the operator [(I-G(s))/s] o M + N is of type M, bounded and coercive. This implies by Brezis [4] or Lions [lo] that it is surjective, hence, (6.16) has a solution us for each s > 0. From (6.16) we obtain 5 5 Ilfll*llu,II, so the coercivity of N shows that {us : s > 0) is bounded in V.

Since N and M are bounded and V is reflexive there is a subset

(which we denote also by {us : s > 0)) for which lim (us) = u, lim N(us) = g and lim M(us) = Mu weakly in the appropriate spaces. * I Let L denote the adjoint of the operator L in V . We apply * (6.16) to any v E D(L ,V) and take the limit to obtain * * I + = , v E D(L ,U). This shows Mu E D(L,V ) and that

(6.17) LMu + g = f

Since N is of type M, it suffices for existence to show

21 5 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

lim sup <_ 0. But from (6.16) and the monotonicity of [(I-G(s))/slMweobtain 5 - I <[(I-G(s))/s]Mu,us-u>. Since Ah E D(L,V ) the left factor of the

last term converges strongly and this gives the desired result.

Uniqueness follows easily from the equivalence of (6.14) and

(6.15); we need only note that A is monotone. QED

Remark 6.10 The extension to nonhomogeneous initial data

is given in Bardos-Brezis [l]with relevant regularity results

on the solution. The problem with periodic conditions Mu(o) =

Mu(T) is solved similarly.

Our next class of results will be obtained from the genera-

tion theory of nonlinear semigroups and its extensions to evolu-

tion equations with multivalued operators. The linear semigroup

theory is inadequate for degenerate problems, but its presentation

above in Section 4 provided an elementary motivation for the

techniques and constructions to follow. Specifically, it indi-

cates the "optimal" hypotheses from which one might expect the

Cauchy problem to be well-posed and it suggests the correct space

in which to look for such results. Also, we shall not present here a nonlinear version of the analytic situation of Theorem 4.2

and Theorem 4.4. * Let E be a vector space with algebraic dual E and let M : * E -+ E be nonnegative (monotone) and self-adjoint. Let K be the

kernel of M and q : E + E/K the corresponding quotient map. The

symetric bilinear function m : E x E + R given by m(x,y) =

21 6 3. DEGENERATE EQUATIONS WITH OEPRATOR COEFFICIENTS

determines a scalar product mo on E/K by

Let W be the Hilkrt space completion of E/K with moy and denote by mo the (extended) scalar product on W. I Let E be the Hilbert space obtained as the strong dual of the space E with the seminorm induced by m. Note that since E/K I I is dense in W we may identify the dual spaces (E/K) = W . Let * I I q : W + E be the continuous dual of q : E -+ W. Since q has * I dense range, q is necessarily injective. Also, each g E E I vanishes on K, hence, g = foq for some f E (E/K) . This shows * * q is surjective, and it follows from (6.18) that q is norm- I preserving. Finally, if Mo : W -f W is the Riesz isomorphism determined by = mo(x.y), then we obtain the identity * (6.19) M = q Moq relating the functions considered above.

Assume we are given for each t E [o,T] a nonempty set I D(t) C E and a (not necessarily linear) function N(t) : D(t) -+ E . Then for each such t we de’fine a multivalued function or relation I No(t) on q[D(t)] x W as the composition

(6.20) No(t) = (q*)-l 0 N(t) 0 q -1 .

Finally, we define the composite relation (t) = Mi’ o No(t on

W x W with domain q[D(t)] for each t E [o,T]. Note that No t)

21 7 SINGULAR AND DEGENERATE CAUCHY PROBLEMS and A(t) are functions if and only if x,y E D(t) and Mx = My imply N(t)x = N(t)y. The following diagram illustrates the various relationships.

N(t) I M E >D(t) ------tE =

We shall consider the semilinear evolution equation

d (6.21) x(MU(t)) + N(t,u(t)) = 0, O<_t

A solution of (6.21) is a function u : [o,T] + E such that I Mu : [o,T] + E is absolutely continuous, hence, differentiable a.e., u(t) E D(t) for all t, and (6.21) is satisfied a.e. on

[o,T]. The Cauchy problem is to find a solution of (G.21) for I which (Mu)(o) is specified in E . We follow the idea from Section 4 of reducing (6.21) to a

“standard” evolution equation with M = identity. If u is a solution of (6.21) and we define v = q 0 u, then (6.19) and (6.20) imply

*I * (6.22) (q MeV) = -N(t,U(t)) E - q No(t,V(t)). * Since q and M are linear isometries, we see that v(t) E q(D(t)) 0 for t E [o,T], v : [o,T] + W is absolutely continuous, and

218 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

I (6.23) v (t) E - A(t,v(t))

is satisfied a.e. on [o,T]. We call such a v(-) a solution of

(6.23). Conversely, suppose v is a solution of (6.23). Then for each t at which (6.23) holds, (6.20) shows that q(u(t)) = v(t) for some u(t) E D(t) and (6.22) holds, By choosing u(t) E D(t) * with q(u(t)) = v(t), hence, Mu(t) = q Mov(t), for all remaining t E [o,T], we obtain a solution u of (6.21). This proves the following.

Lemma 6.11. If v is a solution of (6.23), then for each t E [o,T] there is a u(t) E D(t) such that u is a solution of

(6.21). Conversely, for each solution u of (6.21), the function v = q 0 u is a solution of (6.23).

Corol1ar.y 6.12 Let uo E D(o). There exists a solution v of

(6.23) with v(o) = q(uo) if and only if there exists a solution u of (6.21) with (Mu)(o) = Mu,. There is at most one solution v of (6.23) with v(o) = q(uo) if and only if for every pair of solutions ul, u2 of (6.21) with (Mul)(o) = (Mu2)(o) = Muo it follows that Mul(t) = Mu2(t) for all t E [o,T], hence,

N(t,u,(t)) = N(t,u2(t)) a.e. on [o,Tl.

Results on the Cauchy problem for (6.21) can now be obtained from corresponding results for (6.23). Turning first to the question of uniqueness, we find that a sufficient condition for the mo-seminorm on the difference of two solutions to be nonincreasing is that each A(t) be accretive: if [x,,w,] and [x,,w,]

219 SINGULAR AND DEGENERATE CAUCHY PROBLEMS belong to A(t), then mo(w1-w2.x,-x2) 2 0. The success of the above construction for which (6.21) and (6.23) correspond to one another is reflected in the following.

Lemma 6.13 The relation A(t) is accretive if and only if the function N(t) is monotone.

Proof: Let w1 E A(xl) and w2 E A(x2). Choose u1 ,u2 E D( t) * with x = q(uj) and N(t,u.) = q M w j = 1,2. Then we have j J o jy

(6.24) m0(W1 -w2 ,xl -x2) = so A(t) is accretive if N(t) is monotone. Conversely, if u1,u2 E * D(t) there is a unique pair w1,w2 E W with N(t,u.) = q Mowj, j = J 1,2. Then [q(uj),wj] E A(t) and (6.24) holds. Hence N(t) is monotone if A(t) is accretive. QED

Remark 6.14 N(t) is strictly monotone if and only if it is injective and A(t) is a strictly accretive function.

Theorem 6.15 Let N(t) be monotone and M + N(t) strictly monotone for each t E [o,T]. Then for each uo E D(o) there is at most one solution u(-) of (6.21) with (Mu)(o) = Mu,.

Proof: Since A(t) is accretive for each t by Lemma 6.13, uniqueness holds for (6.23). The result then follows from

Corollary 6.12. QED

We consider now the existence of solutions. Specifically,

220 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS we shall recall the existence results of Crandall and Pazy [l] as they apply to the rather special Hilbert space situation of

(6.23). (The monograph of H. Brezis [3] provides an excellent introduction to this topic with an extensive bibliography; cf. also the recent monograph of Browder [7].! The equation (6.23) has a (unique) solution v with v(o) specified in q(D(o)) under the following hypotheses:

Each A(t) is accretive and I + AA(t) is surjective for all A > 0.

It follows that JA(t) (I+AA(t))-’ is a function defined on all of w.

The domain of A(t), q[D(t)], is independent of t; we denote it by q[D].

There is a monotone g : [o,m) + [o,~) such that

11 JA(t,x)-JA(s,x) 5 A [ t-s[ g( I[xllw)

.(l+inf rllyllw : y E A(s,x)I), t,s 2 0, x E W, O

The preceding results with Corollary 6.12 lead to the following.

Theorem 6.16 Let M be nonnegative and symmetric from the * I linear space E to its dual E . Denote by E the dual of the linear topological space E with the seminorm ~’~; El is a

Hilbert space with norm given by IlfllEl = sup {II : x E E, 5 1). For each t E [o,T] assume we are given a I (possibly nonlinear) N(t) : D(t) + E with domain D(t) C E.

Assume further that for each t, N(t) is monotone and M + AN(t) : I D(t) + E is surjective; M[D(t)] is independent of t; and for

221 SINGULAR AND DEGENERATE CAUCHY PROBLEMS some monotone increasing function g : [o,m) + [o,m) we have

Then for each uo E D(o) there exists a solution u of (6.21) with

(Mu)(o) = Muo.

Proof: From Corollary 6.12 we see that it suffices to show

(6.23) has a solution v with v(o) = q(uo), and this will follow if we verify the hypotheses on CA(t)) above. First note that each A(t) is accretive by Lemma 6.13. Also, M[D(t)] independent of t implies the same for q[D(t)]. To determine the range of * * * I + AA(t) we have (q Mo)(I+AA(t)) = q Mo + ANo(t) = q Mo +

AN(t)q-’ = (M+AN(t))q-l on q[D(t)]. The range of the above is I * E and q Mo is a bijection, so I + AA(t) is necessarily onto W.

The estimate on JA(t,x) follows from (6.25) and we refer to

Showalter [5] for the rather lengthy computation. QED

Remark 6.17 In contrast to preceding results of this section, N need not be coercive or even defined everywhere on V.

Thus it may be applied to examples where N corresponds to a not necessarily elliptic differential operator (cf. Theorem 4.1 and

Example 4.6).

Second order evolution equations with (possibly degenerate) operator coefficients on time derivatives can be resolved by the preceding results. These contain as a special case a first

222 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS order semilinear equation which is nonlinear in the time derivative.

Theorem 6.18 Let A and C be symmetric continuous linear I operators from a reflexive Banach space V into its dual V ; asl I sume C is monotone and A is coercive. Denote by Vc the (Hilbert

space) dual of V with the seminorm '". Let B : V -+ VI be a (possibly nonlinear) monotone and hemicontinuous function. I Then for each pair u1,u2 E V with Aul + B(u2) E Vc there exists 1 I a unique v E L (o,T;V) such that Cv : [o,T] -+ Vc is absolutely d I continuous, &Cv) + B(v) : [o,T] -+ V is (a.e. equal to) an absolutely continuous function, Cv(o) = Cu,,[$Cv)+B(v)](o) = Aul, and

Proof: On the product space E E V x V consider M[x,y] = I [Ax,Cy], hence, El = V' x Vc. Define D = ([x,yl E V x D(B) :

I I Ax + B(y) E Vc) and N : D + E by N[x,y] = [-Ay,Ax+B(y)]. Then apply Theorem 6.16 to obtain existence of a solution u = [w,v] 2 of (6.21). Note that C + AB t h A is monotone, hemicontinuous

I and coercive, hence onto V', and this shows M t AN is surjective

(cf. proof of Theorem 4.3). The second component v of this solution u with Mu = M[u,,u2] isthe solution of (6.26). Uniqueness follows from Corollary 6.12 since M[xl,yl] = M[x2,y2] and

N[xl,~ll = N[x2,y21 imply [xl,yll = [x2,y21. QED

223 SINGULAR AND DEGENERATE CAUCHY PROBEMS

Remark 6.19 The first component of the solution u of (6.21) in the preceding proof is the unique absolutely continuous dw I w : [o,T] -f V with B(a) : [o,T] -f Vc absolutely continuous and I satisfying w(o) = ul, Bw (0) = Bu2,

dw (6.28) + B(a) + Aw(t) = 0, a.e. t E [o,Tl.

Remark 6.20 The special case that results from choosing C =

0 is of interest and should be compared with (6.21). Specifi- cally, (6.26) and (6.27) become first order equations in which the time derivative acts on the nonlinear term. Also, the in- vertibility of A was never used in solving (6.26). It is sufficient for existence of a solution of (6.26) that B + AA be coercive for each A > 0; however, uniqueness may then be lost

(e.g., take A = 0).

We briefly sketch some applications of the results of this section to initial-boundary value problems. Let G be a bounded domain in Rn with smooth boundary aG, Wp(G) be the Sobolev space of u E Lp(G) with each D.u = au/ax. E Lp(G), 1 5 j 5 n, and J J D u = u. Let functions N.(x,y) be given, measurable in x E G 0 J and continuous in y E Rn+’ . Suppose for some C > 0, c > 0, and g E Lq, q = p/(p-l), p L 2, we have

224 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

n + g(x) 2 CIYIP,

n+l for x E G, y,z E R , and 0 5 k 5 n. Let Du = {D.u : 0 j J n}, let V be a subspace of Wp(G) containing Ci(G), and define

I N: V+V by

(6.32) = 1 N.(x,Du(x))D.v(x)dx, u,v E v. j=o 1 G J J

The restriction of Nu to C:(G) is the distribution on G given by - n (6.33) NU = - 1 D.N.(.,Du) + No(*,D$). j=1 J J The divergence- theorem gives the (formal ) Green' s formula = lam v(s)ds, v E V, where the conormal derivative n au on aG is given by -= 1 N.(*.Du)nj and (nly nn) is the a' j=1 . . ., J I unit outward normal. The operator N : V -t V is bounded, hemicontinuous, monotone and coercive; cf. Browder [3, 61, Carroll [14] or Lions [lo] for details and construction of more general operators.

Example 6.21 Let m(-) E Lm(G), mo(x) 2 0, a.e. x E G, and I define M : V -t V by

(6.34) = m(x)u(x)v(x)ds, u,v E v. lG 2 Let uo E V with N(uo) = ml"h for some h E L (G). Then each of

225 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Theorems 6.2, 6.6, and 6.16 gives existence of the solution to the elliptic-parabolic equation - (6.35) D,(m(x)u(x,t)) + N(u(x,t)) = 0 with initial condition m(x)(u(x,o)-uo(x)) = 0, x E G, and a boundary condition depending on V. If V = W:(G), the Dirichlet boundary condition is obtained, while the (nonlinear) Neumann condition au/aN = 0 results when V = Wp(G). The third boundary condition is obtained by adding to (6.32) a boundary integral.

The assumption that m(*) is bounded may be relaxed; cf. Bardos-

Brezis [l]or Showalter [5]. If we add the boundary integral r u(s)v(s)ds to (6.34) and if V is the space of functions in J aG Wp(G) which are constant on aG, we obtain the boundary condition of fourth type (cf. Adler [l])

This problem is solved by Theorems 6.9 and 6.16. By adding appropriate terms to (6.32) and (6.34) involving integrals over portions of the boundary (or over a manifold S of dimension n-1 in c) one obtains solutions of equation (6.35) subject to degenerate parabolic constraints

D,(m(s)u(s,t)) - div(a(s)grad u(s,t))

-- s E s, where the indicated divergence and gradient are in local

226 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS coordinates on S and the coefficients are nonnegative (cf.

Showalter [2, 51).

2 Example 6.22 Choose V = Wo(G) and define nr = Dju(x)D.v(x)dx, u,v E v, j=1 G J

where each c. E Lm(G) is nonnegative, 0 5 j 5 n. Let B = N be J given by (6.32) and assume (6.29), (6.30) with 1 < p 5 2. Then

Theorem 6.18 gives a unique solution of't'?e equation n ." n (6.36) Dt{Dt[cov - 1 Dj(cjDjv)] + N(v)) - 1 D?v = 0, j=l j=1 J with Dirichlet boundary condition and appropriate initial condi-

." tions. Note that N is given by (6.33). Special cases of (6.35) include the wave equation (4.9), the viscoelastici ty equation

(4.10), parabolic equations (4.6), and Sobolev equations with first or second order time derivatives.

Example 6.23 A special case of (6.36) has been of eonsid- erable interest. Specifically, we choose C 5 0, N = 0 for 1 5 j j S n, and NO(x,s) = m(x)IsIP-l sgn (s) where m E Lm(G) is nonnegative and 1 < p I 2. This gives the equation

(6.37) Dt(m(x)Iv(x,t)IP-l sgn v(x,t)) - Av(x,t) = 0.

The change of variable u = IvlP-' sgn(v) puts this in the form

227 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

n (6.38) D,(m(x)u(x,t)) - (4-1) 1 D.()u)q-2Dju(x,t)) = 0 j=1 J with q - 2 = (2-p)/(.p-l) 2 0. This equation arises in certain diffusion processes; it is "doubly-degenerate" since the leading coefficient may vanish and the power of u may vanish independently. Note that the second operator is not monotone so the equation is not in the form of (6.21).

Remark 6.24 Theorem 6.2 is from Brezis [l]. See Lagnese

[lo] for a variation on Theorem 6.6 and corresponding perturbation results on such problems. Theorem 6.9 is contained in the results of Bardos-Brezis [1] along with other types of degenerate evolution equations not covered here (cf. Lagnese [5] for corresponding perturbation results). Theorems 6.15, 6.16 and 6.18 are from Showalter [5]. We refer to Aronson [l],Dubinsky [l], Lions [lo] and Raviart [l]for additional results on (6.38) and, specifically, to Strauss [l]for a direct integration of semilinear equations of the form (cf. (6.37))

(6.39) Dt(N(u)) + A(t)u(t) = f(t) with nonlinear N and time-dependent linear {A(t)) in Banach spaces. Cf. Brezis [l],Kamenomotskaya [l],and Ladyienskaya-

Solonnikov-Uralceva [2] for a reduction of certain (Stefan) free- boundary value problems to the form (6.39), hence, (6.36), and

Lions-Strauss [8] and Strauss [2, 31 for additional results on

(6.28). Crandall [3] and Konishi [l]have studied special cases

228 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

of (6.39) by different methods.

3.7 Doubly nonlinear equations. We briefly consider some

first order evolution equations of the form

(7.1) $(u(t)) t N(u(t)) = 0

where (possibly) both operators are nonlinear. Some sort of de-

generacy is desirable (and necessary for applications) so we

specifically do not make assumptions on M of strong monotonicity

as was done in Theorem 3.12. Moreover, we can include certain

related variational inequalities (e.g., by letting one of

the operators be a subdifferential) so we permit the operators to

be multivalued functions or relations as considered in Section

3.6.

Let B be a real reflexive Banach space, let D(M) and D(N) be

subsets of B and suppose M : D(M) -+ B and N : D(N) -+ B are multivalued functions. That is, M c D(M) x B and N C D(N) x B. (When M and PI are functions, we identify them with their respec-

tive graphs.) We call the pair of functions u,v a solution to

the differential "incl usion"

if u(t) E D(M)fl D(N), v : [o,m) -+ B is Lipschitz continuous, hence, strongly-differentiable a.e., and (7.2) holds. If M and

N are functions, then (7.1) is clearly satisfied a.e. on [o,m), whereas if only M is a function we replace the equality symbol in

229 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(7.1) by an inclusion. Let D(A) M[D(M) fl D(N)] denote the

indicated image in B and define a relation A : D(A) -f B by the 1 composition A = N 0 M- . That is, (x,y) E A if and only if for some z E D(M) fl D(N) we have (z,x) E M and (z,y) E N. Thus, if the pair (u,v) is a solution of (7.2), then it follows that

so we call a Lipschitz function v : [o,~) -f B a solution of (7.3) if v(t) E U(A) and (7.3) holds. Conversely, if v is a solution of (7.3), the definition of A shows there exists for each t L 0 a u(t) E D(M) fl D(N) for which the pair u,v is a solution of

(7.2).

Theorem 7.1 Let M and N be multivalued operators on the real reflexive Banach space B; assume M + N is onto B and that

II (x1-x2)+s(Y1-Y*)11 2 II xp2II Y for s > 0, (7.4)

and (z.,Y.) EN, (z.,x.) E M, j = 1,2. JJ JJ

Then for each uo E D(M)n D(N) and vo E M(uo), there exists a solution pair u,v of (7.2) with v(o) = vo.

Proof: By the preceding remarks t suffices to show (7.3) has a solution. From (7.4) it follows that the sane estimate holds for s > 0 whenever (x.,y.) E A; this is precisely the JJ statement that A is accretive (cf. Crandall-Liggett [2]).

230 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Furthermore, it is easy to check that (x,y) E I + A if and only if for some z E D(M) nD(N) we have (z,y-x) E N and (z,x) E M, so it follows that the range of I + A equals the range of M + N.

Thus I + A is hyperaccretive and it follows from results of

Crandall-Liggett [2] that (7.3) has a unique solution for each vo E D(A). QED

Remark 7.2 If u ,v and u2,v2 are solutions of (7.2), 11 then it follows 11 vl(t)-v2(t) 11 5 Ilv,(o)-v,(o) 11. If vl(o) = v2(o) and either M-l or N-l is a function, then the solution pair u,v is unique.

Remark 7.3 In a Hilbert space H with inner product (*,*),, the condition (7.4) is equivalent to

whenever (z . ,x .) E M (x1-x2Y Y1-YZ)H JJ (7.5)

and (z y ) E N for j = 1,2. jyj

For linear functions this is the right angle condition (cf.

Grabmuller [l],Lagnese [2, 4, 81, and Showalter [6, 14, 15, 16,

201). The difficulty in applying Theorem 7.1 to initial-boundary value problems arises from the assumption (7.4) which relates the operators M and N to each other. A desirable alternative which we shall describe is to place hypotheses on each of the operators independently. The results of Section 3.6 were

231 SINGULAR AND DEGENERATE CAUCHY PROBLEMS obtained when the leading (linear) operator was symnetric. The nonlinear analogue of this is that this operator be a differential or, more generally, a subdifferential. Recall that the subdifferential at x E E of an extended-real-valued lower semi- continuous convex function $ on the locally convex space E is the

I set a$(x) of those u E E verifying

(Cf. Brezis [3], Ekeland-Ternan [l],Lions [lo].) Some examples will be given below. In general, the subgradient a$ is a multi-

I valued operator from (a subset of) E into E .

Theorem 7.4 Let V1 and V2 be real separable reflexive

Banach spaces with V1 C V2, V1 is dense in V2, and assume the inclusion is compact. Let (I and $z be convex continuous (ex- 1 tended) real-valued functions on V1 and V2, respectively. As- sume their subdifferentials N = a$l and M E a$z are bounded (cf. Section 3.6) and that N is "coercive" in the sense that lim inf I$~(U)/IUIP : lulv -+ -1 > o for some p, 1 < p < m. v1 1 Suppose we are given uo E V1, vo E M(uo), and f E Lm(0,T;V;) with df/dt E Lq(0,T;V;) where l/p + l/q = 1. Then there exists a pair of functions u E Lm(o,T;V1), v E Lm(0,T;V;) with dv/dt E

L~(O,T;V;) satisfying

a.e. on [o,T] and v(o) = vo.

232 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Remark 7.5 The compactness assumption above limits application of Theorem 7.4 to parabolic problems; cf. examples below.

We illustrate some applications of the preceding results through some examples of initial-boundary value problems.

Example 7.6 Let V = W;”(G), the closure of Ci(G) in

W1’p(G), p 2 2, and define the nonlinear elliptic operator I

T : V -f V by (6.32), where the functions N.(x,y) satisfy (6.29), J (6.30) and (6.31). For j = 1,2, let m.(*) E Lm(G) be given with J mj(x) 2 0, a.e. x E G, and let k > 0. Set H = L2(G) so V CHc I V , and define D(M) 2 {u E V : mlu + m2T(u) E HI, M(u) mlu + m2T(u); D(N) 2 {u E V : (k/ml(*))Tu E HI, N(u) :(k/ml(*))T(u).

Then M and N are (single-valued) nonlinear operators in the

Hilbert space H. To apply Theorem 7.1, we first verify (7.4) in its equivalent form (7.5). For u,v E D(M) fl D(N) we have (MU-MV~NU-NV)~= k + I ( km2(x)/ml (x)) (T(u)-T( v) G dx 2 0, so (7.4) holds. To show M + N is onto H, let w E H and consider the equation

The coefficients all belong to Lm(G), so the operator on the left side of (7.7) is monotone, hemicontinuous and coercive from V

I to V , hence, surjective, so there is a solution u E V of (7.7).

Since m,(*) and m2(-) are bounded, it follows from (7.7) that u E D(M)n D(N) and that M(u) + N(u) = w. Thus, Theorem 7.1 gives existence of a weak solution of an appropriate initial-

233 SINGULAR AND DEGENERATE CAUCHY PROBLEMS boundary value problem containing the equation

where T[u] is given by (6.33). Flote that the coefficients m1 and m2 may vanish over certain subsets of G.

Example 7.7 We apply Theorem 7.4 with V1 = WAYp(G) and V2 = n La(G), ct 5 P, Q1(u) = l/p 1 IDju(X)IPdx, @,(u) = G j=l l/ct lu(x)j"dx, and appropriate f : [o,T] -+ W-lgq(G). The cor- IG responding subdifferentials are given by 34, (u) =

we obtain existence of a solution of an initial-boundary value problem for the equation

Example 7.8 Let V1,V2, @1 and f be given as above but

-c define @,(u) = l/a iGlu + (x)l"dx, u E V2, where u (x) = 0 when u(x) < 0 and u+(x) = u(x) when u(x) >, 0. Theorem 7.4 gives existence for an initial-boundary value problem for

Example 7.9 Let V2 be a Hilbert space and set @,(u) = 2 max {1/2,1ulH/21. The subdifferential is given by

lulH < '9 IuI = 1, lulH '

234 3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

If v1 and N are given as in Theorem 7.4, there is then a solution pair u,v which satisfies the abstract "parabolic" equation I -u (t) + f(t) E N(u(t)) where lu(t)l > 1 and the "elliptic" equation f(t) E N(u(t)) where lu(t)l < l.

Remark 7.10 Theorem 7.1 is an extension of the corresponding result of Showalter [4] when M and N are functions. Theorem

7.4 and Examples 7.7, 7.8 and 7.9 are from Grange-Mignot [l].

See Barbu [l]for related abstract results and examples where M and N are subdifferentials which are assumed related by an assumption similar to (7.4) but distinct from it. The equation

(7.9) was studied directly in Raviart [2]; cf. Chapter IV.1.3 of Lions [lo]. A doubly-nonl inear parabolic-hyperbol ic system is discussed by Volpert-Hudjaev [l],and parabolic systems are considered by Cannon-Ford-Lair [3]. These last two references indicate how certain doubly-nonl inear equations arise in applications; for applications of equations of the type (7.8), cf.

Gajewski-Zachari us [2].

235 This page intentionally left blank Chapter 4

Selected Topics

4.1 Introduction. In this chapter we will discuss

briefly some further questions arising in the study of singular or degenerate equations. Some of the presentation will be in a more "classical" spirit and we will follow the original papers

in describing problems and results. Proofs will often only be

sketched or even omitted entirely, mainly for reasons of space and time, but suitable references will be provided (without at-

tempting to be exhaustive in this respect); further relevant information can often be obtained from the bibliographies in these

references. The material considered by no means covers all possible questions and has been selected basically to be represen-

tative of the historical development and to reflect some current

research trends in the area. Thus we will deal for example with

Huygens' principle, radiation problems, initial-boundary value problems, nonwellposed problems, etc., among other questions;

the section title will indicate the problem under consideration.

4.2 Huygens' principle. Referring back to (1.4.8) and (1.4.10) with A = -A let us suppose m t p =; - 1 with T a X function so that wm+P(t) is given in terms of a surface mean value ux(t) * T of T(<) over the sphere [x-

237 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

I x-s = t, as was noticed by Diaz-Weinberger [2] and Weinstein 141. To picture this one can draw the retrograde "light" cone from (x,t) E Rn x R+ to the singular hyperplane t = 0 and look at the intersection. We note here that for the wave equation where m = - 1/2,n = 2p + 1 is required above which reproduces a well known fact. This phenomenon furnishes an example of what is known as the minor premise in Huygens' principle. In general Huygens' principle can be enunciated in various versions involving other features as well (cf. Baker-Copson [ll , Courant-Hi1 bert 111, Hadamard [l], Lax-Phillips [l]) but we shall simply say here that a linear second order partial differential operator L(Dx ,Dt) (k = 1, ...,n) is of Huygens' type in a region R C k Rn xR if the value u(x,t) of the solution of Lu = 0 at a point

(x,t) E R depends only upon the Cauchy data on the intersection of the space like initial manifold with the surface of the characteristic conoid with vertex (x,t). This formulation and others play an important role in wave propagation for example and the matter has been studied extensively. (In addi tion to the above references we cite here only some more or less recent work by Bureau [l], Douglis [l], Fox [l], Giinther [l; 2; 3; 4; 51, Helgason [4], Lagnese [ll; 12; 131, Lagnese-Stellmacher [141, Solomon [l; 21, and Stellmacher [l].) In dealing with Huygens' principle for singular problems we will follow Solomon [l; 21 and only make a few remarks about other work on singular problems later. Thus let us consider the operator

238 4. SELECTED TOPICS

(2.1 1 E"' = a2/at2 + a/at + c(t) where Rem > - 1/2 or m = - 1/2 and c(*) is a complex valued function belonging to L'(o,b) (O

m (2.2) Emum = Au ; um(0) = T E D'; uF(0) = 0

1m where A is the Laplace operator in Rn and since c (0) E L , u is required only to satisfy the differential equation almost everywhere (a.e.). Solomon [l;23 employs the Fourier technique developed in Chapter 1 and looks for resolvants Z;(t) satisfying "2 (2.2) with T = 6. Setting A2 = 1 yk = lyI2 one then looks for a 1 function im(y,t) = FxZ:(t) satisfying

"m (2.3) Em^Zm + A2^Zm = 0; Z (y,o) = 1; iy(y,o) = 0

A solution ?(y.t) of (2.3) is obtained by converting (2.3) into an integral equation as in Chapter 1 (cf. (1.3.24)) and using a

Neumann series such as (1.3.25) (cf. also Section 1.5). Using techniques similar to those of Chapter 1 the solution is obtained in the following form (see Solomon [l],pp. 226-231).

Theorem 2.1 The unique solution of (2.3) for t E [o,b] and y E Rn can be written as

(2.4) "mZ (y,t) = im(y,t) + im(y,t) where imis given by (1.3.6) with z = tlyl and im is a "smooth"

239 SINGULAR AND DEGENERATE CAUCHY PROBLEMS perturbation satisfying the estimate

for IyI > A0 and o S t 5 b (ko and ho are independent of (y,t)).

The corresponding unique resolvant Z!(t) E El has support contained in the ball 1x1 5 t and um(t) = Z!(t) * T is the unique I solution of (2.2) in D .

Remark 2.2 The estimate (2.5) is useful in determining the order of the distribution Z!(t); in particular for Rem + 1/2 > n - 1 it follows that Z!(t) will be continuous in x for 0 < t i b.

We will say now that (2.2) is of Huygens' type if and only if the support of Z!(t) is concentrated on the sphere 1x1 = t.

Note here that this definition depends on the choice of initial data when m = - 1/2 and n = 1 for example since the one dimensional wave operator is not a Huygens' operator in the sense previously delimited for arbitrary Cauchy data u(x,o) and 22 ut(x,o). Solomon uses the distributions 6(')(\xl -t 1, v =

V 0, 1, 2, ..., defined for t > 0 and n > 1 by (cf. Gelfand-Silov [51)

Here 1x1 = p, 8 denotes the n - 1 angular -variables (el, ..., in polar spherical coordinates with +(p,B) = +(x), and Rn is the surface of the unit sphere in Rn. For n = 1 one writes

240 4. SELECTED TOPICS

V (2.7) dV)(1 x = L[&wax) [6(x+t) + 6(x-t)] 21tl 22 clearly the distr butions &'((XI -t ) are concentrated on the sphere 1x1 = t. Then the main result in Solomon [ll is given by

Theorem 2.3 A necessary and sufficient condition for (2.2) to be of Huygens ' type is that n - (2m+l) = 2N + 1 with N L 0 an integer while there exists a resolvant Z:(t) of the form

where c, = (-1 )Nn-n'2r(m+l), a. = 1 , and for v = 1 , . . . , N (if

N > 0) one has E-m(aN) = 0 with

(2.9) 4(ta\: + vav) = E-m(av-l)

V Remark 2.4 We recall from Gelfand-Silov [51 that for v = 0, 1, 2, . . .

(2.10) FS(')( 1x12-t2) = ;(v,n)lyl 2v-n+221/2(n-2)-vJ 1 (2) -(2 n-2) -v where c(v,n) = 7(-1/2)v(2n)n/21 and z = tlyl. Then examples to

illustrate Theorem 2.3 can be given as follows. Suppose first

that Z:(t), expressed by (2.8), has precisely one term (i.e.,

a = 0 for v _> 1). Then by (2.9) E-m(ao) = c(t) = 0 since a. = V 1 and we are in the EPD situation. Let N be the integer involved

in (2.8) with of necessity n - (2m+l) = 2N + 1 an odd positive integer as indicated. To check the form of Z:(t) given by (2.8)

in this case we use (2.10) to obtain Zm(y,t) = cmt-ZmaoFG(N)

241 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

22 1 Am (1x1 -t ) and since $n-2) - N = m this yields Z (y,t) = "m 2mr(m+l)z-mJm(z) = R (y,t) as required by (1.3.6). Next suppose that (2.8) has no more than two terms (i.e., av = 0 for v _> 2).

From (2.9) and the requirement E-m(al) = 0 we have ta; + al = E-m(ao) = c(t) or c(t) = (ta,)' and E-m(al) = a;' + +;1 2m) t I' c(t) al = 0. Combining these two equations one has al + 9

2mal + -t2 = ct for cto any constant. Making a change of vari- 2 al 0 ct 2' " 2m+l ' o ables al(t) = w /w this becomes w - 7 - -2 w = 0. 2 Now in terms of w we have al(t) = (log w)' with c(t) = 2 (log w)" and suitable w can be found as follows. If cto = 0 we get w(t) = cr + f3t2m+2where a and B are arbitrary constants but may depend on m. In order that the resulting a, be such that (2.8) satisfies the resolvant initial conditions we must have c1 j o while for m = - 1/2 it is necessary that B = 0. If cto f 0 the solutions w of the differential equation for w will be of the k form w = exp (42) F(- ;T; - k; T) where T = tal/', k = 2m+l , and

F (y; 6; t) is any solution of Kunnner's confluent hypergeometric equation tF" + (6 - t) F' + yF = 0. Upon examination of the behavior of such F(y;6; t) as t + o it follows that for the resulting a, to satisfy the appropriate initial conditions F must be k k taken in the form F = alF1 (- 2; - k; T) + f3 F2 (- 2; - k; T) where F2 is any solution of the Kumner equation independent of lF1y a f 0, and when m = - 1/2 (i.e. k = 0) 6 = 0. Further observations can be found in Solomon [1;21.

242 4. SELECTEDTOPICS

Remark 2.5 We will deal here explicitly only with some work of Fox [l]on the solution and Huygens' principle for singular

Cauchy problems of the type

with data um(x,o) = T(x) and u;(x,o) = 0 given on the singular hyperplane t = 0. Except for some results of Gunther (loc. cit.) most of the other work on Huygens' principle mentioned earlier deals with nonsingular Cauchy problems so we will not discuss it here; Gunther's work on the other hand is expressed in a more geometric language which requires some background not assumed or developed in this book and hence the details will be omitted. 2n 2 Thus let r = r (x,t; C) = t - 1 (xk - Ck) be the square of the k= 1 Lorentzian distance between a point (x,t) E Rn x R and a point

(5,o) in the singular hyperplane t = 0. We denote by (2.11)-,, the equation (2.11) with index -m and observe that if u-~satis- t-2m -m fies (2.11)-,, then um = u satisfies (2.11) (cf. Remark

1 1.4.7). Fox composes the transformations zk = /4xkCk and yk = ( 5k/xk) r (k = 1, . . ., n) to produce new variables (z,r,y) in place of (x,t,C) (suitable regions are of course delineated). A m- -n solution of (2.11)-,, is sought in the form u-" = r 2 V(z) where V(z) is to be determined. After some computation one shows that such ummare solutions if V(z) satisfies a certain system of n ordinary differential equations and such a V(z) can be found in the form V = FB where FB is the Lauricella function given for

243 SINGULAR AND DEGENERATE CAUCHY PROBLEMS lzkl < 1 by (cf. Appell - Kamp6 de F6riet 111)

n (2.12) FB = FB (a, 1-a, m - + 1, z)

11 where ak = + (1-4Ak)'/2 and (8,s) = r(R+q)/I'(R). The series defining FB converges uniformly in (a,m,z) on closed bounded n regions where lzkl < 1 and m - + 1 # 0, -1, -2, . . . . One notes that since ak and 1 - ak are complex conjugates for Xk real, the products (ak,pk) (l-akypk) are real, and thus FB s real when n m and the Ak are real. If Rem > 2-1 and lxkl > It1 > 0 it makes sense then to consider as a possible solution o the singular Cauchy problem for (2.11) the function.

m (2.13) uX(x,t,T) = k,

n where km = r(mc1)/vn/2r(m-"+l)2 and v!(x,t,C) = Itl-2mrm-T FB is a solution of (2.11). We assume here that T(-) is suitably differentiable and denote by S(x,t) the region of the singular hyperplane t = 0 cut out by the characteristic conoid with vertex (x,t) (thus S(x,t) is an n-dimensional sphere of radius

It1 and center x in this hyperplane determined byr(x,t,S) 2 0).

As described, with lxkl > It1 2 0, S(x,t) does not intersect any of the hyperplanes xk = 0 and lies in an octant of the hyperplane t = 0. Note here also that the factor r(m-;+l)-l in k, can be n S played off against (m -?+ 1, Ipk)-' in FB to remove any dif- n 1 ficulties when m- t 1 = -p. Now for Rem > 5 + 1 one checks

244 4. SELECTEDTOPICS

easily that Lyu: = km T(<)L!)r(x,t,<)d< = 0 when T (0) is

continuous, since all boundary integrals which arise will vanish.

Using analytic continuation and after several pages of interest-

ing analysis Fox shows that (for m # -1 ¶ -2, . . .) if T(-) E n c{2+p) where p is the smallest integer such that Rem 2 2 - p -l\ m then uAYdefined by (2.13), satisfies the singular Cauchy prob-

lem for (2.11) ({2+p) = 0 if 2 + p i 0). Uniqueness of the

solution uy(x,t,T) is proved for 2m + 1 2 0 and it is shown that Huygens' principle holds when, for some nonnegative integer p,

the parameters m and xk satisfy m = - p - 1, xk = ak(l-ak) n where ak E ~ly...yptl~,and 1 ak i n + p. For m # -1, -2, ... 1 the solution u:, although not unique for Rem < - 1/2, is the unique analytic continuation of the unique solutions uy with

2m + 1 2 0 and the criteria above for Huygens' principle remain valid (in fact necessary and sufficient).

4.3 The generalized radiation problem of Weinstein. One

can read extensively on the subject of radiation for the wave

equation and we mention for example Courant-Hilbert [ll Lax-

Phil lips [l]¶ Somnerfeld [l]¶ and references there. The usual

formulation (cf. Weinstein (61) prescribes a function f(*) with

f(t) = 0 for t < 0 and asks for a function u satisfying utt =

Au inRP xIR with u(x,o) = ut(xYo) = 0 and such that

(3.1) lim dun = - wnf(t) r4 2 n2 where r = 1x1' = 1 xi¶ w n = 2~~/~/r(n/2),and theintegral in 1

245 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(3.1) is over a sphere of radius r inlRn. When n = 3 for example one has the well-known solution u(x,t) = f(t-r)/r and one is led to look for solutions which depend only on t and r. This entails a radial form of A and yields a singular problem in two vari abl es

n-1 u= +- (3.2) tt 'rr r 'r with u = 0 for r 2 t and for r = 0, u should have a singularity of the form r-(n-2). Now replace n - 1 by 2m + 1 and consider a generalized radiation problem

= m 2m 1 (3.3) +

tt 'rr + -r r where we recall that if urn satisfies (3.3) with index m then uqm = r2mum satisfies (3.3) with index -m (cf. Remark 1.4.7).

If 2m + 1 = n - 1 then 2m = n - 2 so that a solution u-~finite near r = 0 corresponds to a solution u" w th a singularity of the desired type as r .+ 0. Thus, thinkin of suitably negative values of Rem (see Theorem 3.2 for the precise range Rem < 0), one looks for a solution um(t,r) of (3.3) satisfying

(3.4) urn(t,o) = f(t); u"(t,t) = 0 where f(t) = 0 for t < 0. This type of problem has been studied in particular by Diaz and Young [l;81 , Lieberstein [l;21, Lions [4], Suschowk [l],Weinstein [6], and Young [6; 121.

We will follow Lions [4] here in treating the generalized

246 4. SELECTEDTOPICS

radiation problem because of the generality of his approach and

the use of transmutation operators, which serves to augment

Section 1.7. First, as a sketch of the procedure, we consider for Rem > - 1/2

m m 2 (3.5) v (t,r) = ~nr m+1/2 z(z2-r2)m-1/2 um(t,z)dz

Then, if u" satisfies (3.3) with um(t,r) = 0 for t 5 r, it m follows that vTt = vrr with vm(t,r) = 0 for t 5 r. Consequently

vm(t,r) = Fm(t-r) where Fm(s) = 0 for s i 0. Now (cf. below)

(3.5) can be inverted in the form

For Rem < - 1/2 (3.6) is valid in the usual sense while both

(3.5) and (3.6) can be extended by analytic continuation to

every m E C. Putting Fm(t-z) in (3.6) and integrating by parts one obtains

For Rem L 0 such um(t,r) do not converge when r + 0 unless Fm, and hence urn, is identically zero. For Rem < 0, on the other

hand, as r + 0

(3.8) um (t,r) +.r(-m+1/2)1 k-2m-1(Fm)l(t-z)dz

so that, recalling the definition of the R - L integral in Section 1.6 (cf. (1.6.20) for example),

247 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

m (3.3) u (t,o) = I-2m(Fm)' = f(t)

-2m-1 m is required. Since I-2m(Fm)' = I (F ) we have Fm determined

uniquely by Fm = (r(-m+l/2)/r(-2m)) 12m+1(f)and putting (3.9)

in (3.7) one obtains

m (3.10) um (t,r) = 1 -m-1/2(12mf)(t-z)dz which is equivalent to Weinstein's solution (cf. Weinstein [61) and demonstrates uniqueness a'l so.

In Lions 141 the formal calculations above are rephrased and justified in terms of transmutation operators as follows.

(We recall from Section 1.7 that B transmutes P into Q if QB =

BP.) Now Let R = (0, m) and D:(R) = D+(R)' be the space of distributions on R with support limited to the right (cf.

Schwartz [l]and note that D+(R) signifies Cm functions with

not necessarily compact support but equal to zero near the origin). For simp1 icity we will occasiona ly use a function

notation for distributions in what follows in writing for example . Then for T E set MT(S) =

x 2 )> for @ E D(RE); T(<) which means that

it is easy to see that M is an isomorphism $(fix) +. D'(R5)

(or DL(Rx) +. $(Q )) with inverse M-lS(x) = S(x 2 ) (i.e., -5 =

calculation yields MfT = (4 5 I? + 2D ) MT for T E D'(Rx) (here

5 5 I Dx = d/dx, etc.). Now referring to the standard spaces D ,

248 4. SELECTEDTOPICS

Di, and D: over R as in Schwartz [ll, we set for p E E, Y = P (l/r(p)) Pf(xP-’)lx>b E Di (the notation Pf is explained in

Schwartz [l]). Then Y = o for x < 0, Y-R = $.a for R a non- P I negative integer, p -+ Y : Q: -+ D+ is analytic and Y P P*yq= I . Set now Z (x) = Y (-x) so that Z E D 9 Z = 0 for x > 0, yP+q P P P -P Z-, = (-I)‘#& for R a nonnegative integer, etc.; in particular

DZ = -Zp-l and XZ = -PZ~+~.Finally one notes that T -+ S*T is P P I I a continuous linear map Di(n) -+ D-(fi) when S E D- is zero for m x > 0 (formally (S*T) (x) = 1 S(x-y)T(y)dy) and we denote by * JX Zp E L (D:(a), DL(S2)) the map T -f Zp*T. Now let Lm = DE t @-$ Dx (L, = Li in Chapter 1) and note

I I that L, : D-(n) -+ D (S2) is continuous. For m E Q: Lions con- 1* structs a transmutation operator H, = M- 0 M of D2 O z-ln-1/2 I 2 into Lm on D (a) (i.e., H,D = LmHm) such that Hm:

I I D-(O) -+ D (Q) is an isomorphism and m -+ H, is an entire analytic

function with values in L(Dl(S2), Dl(0)). The inverse H, = Hi’ = * 2 v-’ 0 zm+1/2 0 M, which necessarily satisfies D H, = HmL, on D - (a), enjoys similar properties and in function notation one can write

(3.11) ff,T(x) = mxr~(y~-x~)-~-~/~T(y)dy; Rem < - 1/2

m 2 (3.12) HmT(x) = I:(yZ-x2)m-1/2r(r)dy; Rem > - 1/2

For other values of m E $ one extends these formulas by analytic

continuation. It is important to note that if T E Dl(n) is

249 SINGULAR AND DEGENERATE CAUCHY PROBLEMS zero for x > aT then HmT = HmT = 0 for x > aT. Lions phrases the generalized radiation problem as follows.

Problem 3.1 Let P cR x R be the open half plane (t,r) where r > 0 and Q c P be the set Q = {(t,r) E P; t 2 r). For

Rem < 0 find urn E Dl(P) satisfying (3.3) with supp umc Q and um(-,r) + f in DI as r + 0 where f E D: is given with f = 0 for t < 0.

The problem is restricted to Rem < 0 since it is shown not to be meaningful for other m E 4 (cf. Lions [4]). There results

Theorem 3.2 Problem 3.1 has a unique solution urn depending

I I continuously in D (P) 0n.f E D, (f as described).

Formally the proof goes as follows. One writes vm =

(1 @Hm)um so that vm is given by (3.5) for Rem > - 1/2 say m (cf. (3.12)). Applying 1 @ H, to (3.3) one has vtt = vrr and v" E D'(P) has its support in Q. Consequently vm(t,r) = Fm(t-r) as before with Fm E DI and Fm(s) = 0 for s < 0. Since H, is an m isomorphism with Hi' = Hm we have u = (1 C9 Hm)vm and for Rem < - 1/2 say we obtain formally by (3.11) (cf. (3.6)) where = (Z/r(-m-l/Z)) i: ~(z~-r~)-~-~/~@(z)dzfor r > 0, Rem < - 1/2, and @ E D-. The function m -+ Si can in fact be extended by analytic continuation to be entire with values in

250 4. SELECTED TOPICS

Di and, for Rem < 0, asr+O, Sk + (Zr(-2m-l)/r(-m-l/Z)) Y-2m-l

I in D,. Hence from (3.13) as r + 0

and this limit must equal f (cf. (3.9)). Consequently the explicit solution of Problem 3.1 for Rem < 0 is

(3.15) um(t,r) which can be rewr tten in a form equivalent to (3.10) (cf Lions [41).

4.4 Improperly posed problems. Improperly posed problems such as the Dirichlet problem for the wave equation or the

Cauchy problem for the Laplace equation have been studied for some time and have realistic applications in physics. For a nonexhaustive list of references see e. g., Abdul-Latif-Diaz

[lI, Agmon-Nirenberg [31, Bourgin [ll, Bourgin-Duffin [21, Brezis-Goldstein [5], Dunninger-Zachmanoglou [21, Fox-Pucci [21 , John [23, Lavrentiev [Z], Levine [l; 5; 10; 111, Levine-Murray [121, Ogawa [1;23, Pucci [21 , Sigillito 141 and a recent survey by Payne [31 (cf. also - Symposium on nonwellposed problems and logarithmic convexity, Springer, Lecture notes in mathematics,

Vol. 316, 1973). In particular there has been considerable investigation of improperly posed singular problems by Diaz-

Young [lo] Dunninger-Levine [lI , Dunninger-Weinacht [31

Travis [ll, Young [4; 5; 7; 8; 113, etc. and we will discuss

251 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

some of this latter work.

The most general abstract results seem to appear in

Dunninger-Levine [l]and we will sketch their technique first.

Let E be a real Banach space and A a closed densely defined

linear operator in E (as is well-known the use of spaces E over R instead of involves no loss of generality). We shall denote

by cs (A) the point spectrum of A. P

Definition 4.1 The E valued function urn(-) is said here to be a strong solution of

on (o,b) if urn(-) is norm continuous on (o,b), urn(*) E C2(E)

I I weakly, and for any e E E

m I 2m+1 rn I m' = 0 (4.2) + -t + ,

m I d2 on (o,b), where = - am,el>, etc., and writing dt2 p(t) = t2m+111 Aum(t)ll for m > - 1/2 with p(t) = t [I Aum(t)ll 1 for m < - 1/2, one has p(*) E L (o,b).

The nonsingular case m = 1/2 is encompassed in forthcoming work of Dunninger-Levine and will not be discussed here. Set

now p = Iml and let (An) denote the positive roots of

Jp(% b) = 0 for m 2 0 or m < - 1/2; if - 1/2 < m < 0 let

{An) denote the positive roots of J b) = 0. Then there -P (6 results

252 4. SELECTEDTOPICS

Theorem 4.2 Let um be a strong solution of (4.1) as in

Definition 4.1 with um(b) = 0. If m > - 1/2 assume t2m+1( 11 uyll + tlIumII)-+ 0 as t -+ 0 while for m < - 1/2 assume that tlI uyll + Ilum 11 -+ 0 as t + 0. Then urn :0 if and only if the sequence {An} satisfies An IS (A) for all n. 4 P The proof can be sketched as follows. Consider the eigenvalue probl em

(4.3) $(b) = 0; t1I2$(t) bounded and take for m 2 0 or m < - /2 the so utions

(4.5) $n,p (t) = Jp(%t, =o while for - 1/2 < m < 0 choose the solutions

(t) = J (.Ant); J (qb) = 0 (4.6) $n,-p -P -P

Assume first that, for some n, An E CJ (A) with Avn = hnvn and P set

tmmJp(Ant)vn; m 2 0 or m < - 1/2 (4.7) urn(t) = t-mJ-p(Jj;nt)vn; -1/2 < m < 0

Such um are strong solutions of (4.1) in the sense of Definition

4.1 and their behavior as t -+ 0 is correct, as specified in the hypotheses of Theorem 4.2. Hence An E u (A) for some n implies P the existence of nontrivial um satisfying the hypotheses of

253 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Theorem 4.2 which means that urn f 0 in Theorem 4.2 implies

An 4 u (A) for all n. P On the other hand suppose An 4 u (A) for all n and denote P by J,n the appropriate functions I$,,, or from (4.5) - (4.6). Suppose urn is a strong solution of (4.1) satisfying the condi- tiosn of Theorem 4.2 and for some 6 E (o,b) set

b (4.8) Vp) = tW1J,,,( t)um(t)dt 6

The u:(6) are we1 defined by virtue of the hypotheses on u"

and one notes eas ly that, as t -f 0, tm1J,,(t) = 0(t2"+') for 1 m > -while t"+'Qn(t) = O(t) for m < Hence t -f - 2 - p+l 1 qJn(t)llum(t) 11 E L (o,b) and vf: = lim $(ti) exists in the norm topology of E as ,6 -f 0. Since A is closed it can be passed under the (Riemann type) integral sign in (4.8) so that

vf:(6) E D(A) and furthermore y: = lim A v:(6) exists as 6 -f 0 under our hypotheses on p(t) in Definition 4.1 with Av: = yf: by I I the closedness of A. For e E E fixed a routine calculation yields now b (4.9) = tmtlqJn(t)dt 6

+ 'n ~~tm''Qn(t)dt

254 4. SELECTEDTOPICS

m1 and in view of the hypotheses one obtains as 6 -+ 0 = n m I m1 ma lim h = h . Therefore <(A-hn)vn,e > = 0 for nn n m any el E El which implies AvY = Anvn which by our assumptions means v: = 0. But one knows that the Bessel functions qn form a complete orthogonal set on (o,b) relative to the weight func- m1 tion t(cf. Titchmarsh [3]) and hence from =

Ibtm’$,,(t) 0 = 0 for any el E El from which results um(t) = 0.

Other boundary conditions of the type uF(b) t aum(b) = 0 are also considered in Dunninger-Levine [l]and uniqueness for weaker type solutions is also treated. One notes that very little is required of the operator A in the above result. The technique does not immediately extend to more general locally convex spaces E because of the norm conditions.

We go now to some more concrete problems which are related to Theorem 4.2 in an obvious way. First we sketch some results of Young [4] on uniqueness of solutions for the Dirichlet problem relative to the singular hyperbolic operator defined by 2m+l ij ij L,u = u + --+A (a u~.)~t cu where c and the a tt - 1 1 tt 11 1 .i are suitably smooth functions of x (xly . . ., xn) and m is real. Let R c R be bounded and open, Q =fi x(o,b), c(x) 2 0 ij for x E R, and assume the matrix (a ) is symmetric and positive definite; sufficient smoothness of the boundary r of R will be taken for granted.

255 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Theorem 4.3 Let Lmum = 0 in Q with um = 0 on rx[o,b) and m 1 2 u (x,o) = 0. Suppose m > - and assume urn E C (Q) 0 C1(q). Then um = 0 in q.

The proof uses the fact that any solution um of Lmum = 0 2 1 belonging to C for t > 0 and to C for t 2 0 necessarily satis- 1 fies uF(x,o) = o for any m + - 2 (cf. FOX [I], Walter [I]). NOW integrate the identity

(4.10) 2u L u = (ut2 + 1 a iju ux +cu')~ + 2 2m+l 2 tm X Wt i,j ij

over the region Qs = Qx(o,s), s 5 by and use the divergence theorem to obtain, when Lmu = 0,

(4.11) [(up.l.aiJux.ux +cu2 )vt-2 a iju u v.lda 1 xi t J IaQs ~YJ 1j i,j + 2(2m+l)I ptdxdt12 = 0 QS where v = (v , . . ., vn,vt) is the exterior normal on aQs = boundary Qs. Putting our urn in (4.11) and using the boundary conditions plus u"'(x,o) = 0 we have the equation [(u:)~ + t J R 1 aijumxi xj + c(um2 ) ]l:t,sdx + 2(2m1) I, l(ut) dxdt = 0. Since 1 (aiJ) is positive definite, c 2 0, and m > - -2 both of these integrals are zero and by continuity the first term then vanishes for 0 5 s 5 b. Hence urn s 0 in 0 since um(x,o) = 0.

1 2 Theorem 4.4 If m 5 - 7 then any solution urn E C (Q) fl m C'(0) of Lmum = 0, with u = 0 on the boundary of Q, vanishes

256 4. SELECTED TOPICS

identically if and only if JLm(%b) f 0, where the An are the nonzero eigenvalues of the problem Av = - z.(a ij vxi xJ t cv in iYJ $2 with v = 0 on r. There is an obvious relation between Theorems 4.4 and 4.2 where A corresponds to the operator determined by Aw = cw - 2 - 1 (aijwx.)x in E = L (L?) with w = 0 on r. The proof of i,j ij Theorem 4.4 goes as follows. First suppose there is a nonzero eigenvalue An such that J-m(Q) = 0 with corresponding eigenfunction vn(x). Then um(x,t) = t-mJ-m(~t)vn(x) sat'isfies

Lmum = 0 and the specified boundary conditions. Conversely if

J-,,(%b) 0 for any An one integrates the identity wLmu - uMmw = (utw-uwt+--Uw)2m+l - 1 [la ij(U~~W-UW~~)]~~ Over the cy- ji 1 b linder Qs enclosed by Q between the planes t = s and t = b (0 < s < b). Here Mm is the formal adjoint of L, given by Mmw = wtt - (2mtl)(w/t)t - l(aijw ) + cw. By the divergence theorem one 'j obtains

(4.12) jQ:( wLmu-uMmw) dxdt

ij = b[(utw-uwt -+ 2mt1 w)vt-la (u w-uwx )v .Ida 'i iJ aQS t"

Now let Lmum = 0 with urn = 0 on the boundary of Q and set wm(x,t) = tmt1J-,(%t)vn(x) where An is a nonzero eigenvalue

of the problem indicated in the statement of Theorem 4.4. It

is easily checked that Mmwm = 0 so that the left hand side in

(4.12) vanishes and since wm = 0 on r we have

257 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

m lm Now as s + 0 wt and p are bounded with um(x,o) = 0 (and m 1 ut(x,o) = 0 for m - 2); the evaluationin(4.13) at t = s tends to zero and we obtain

since um(x,b) = 0. Consequently for n = 1, 2, . . . uy(x,b)vn(x)dx = 0 and by the completeness of the vn(cf. R Hellwig [3]) it follows that uF(x,b) = 0. Now one may integrate b m m (4.10) over Qs and since u (x,b) = ut(x,b) = 0 there results

1 -(um2 ) dxdt = 2(2m+1)jQbt t S 1 Since it follows that for any s, 0 5 s 5 b, m 5 - -2 +c(um)']/ dx = 0 and the conclusion of the t=s theorem is immediate.

Young also treats the Neumann problem by similar techniques. Travis [l]uses a different method with more general boundary conditions and breaks up the range of m into parts corresponding the the Weyl limit circle and limit point situations (cf. Coddington-Levinson [l]). Thus one considers the singular eigen va 1ue problem

258 4. SELECTEDTOPICS

,Itzm+’I$( t)I ‘dt < m; $(b) = 0

It can be shown that for m 5 -1 or m L 1 the problem (4.16) has a pure point spectrum and the eigenfunctions form a complete 2m+l orthogonal set on (o,b) relative to the weight function t . Furthermore if f has finite tZm+’norm on (o,b) (i.e.¶ b $m+l 2 [ If(t)l dt < m) then the Fredholm alternative (cf. Hellwig ’0 2m+l $’ I + [3]) holds for the nonhomogeneous equation (t Atzm+’$ = tZm+’f. In the limit circle case - 1 < m < 1 there is always a pure point spectrum with a complete orthogonal set of eigenfunctions $n(relative to the weight function tZm+’)for the problem

2m+l$l)’ + t2rn+lA$ = (4.17) (t 0; $(b) = 0;

I I where W($,J,,t) = $J, - $ J, and J, = +(tyAo) is a possibly complex valued solution of the different a1 equation n (4.17) for X = 2m+ 1 ho satisfying t W($¶Tyt) + 0 as t -f 0 (such J, exist). The

Fredholm a1 ternative also holds again for the corresponding nonhomogeneous problem. 1 We take now aij(*) E C in a bounded open set R c Rn with ij sufficiently smooth boundary r where aiJ = aji and Fa Cisj 2 u1El2 while c(=) is continuous in R. Let a/av denote the

259 SINGULAR AND DEGENERATE CAUCHY PROBLEMS transverse derivative a/av = laij(x) cos (v,x.) a/ax. where v J 1 2m+l is the exterior normal to r and let Lmu = utt + -pt - l(aiju ) + cu as before. x.1 'j

Theorem 4.5 Let m 4 (-1,l) and Lmum = 0 in 3 x (o,b) with u" E C2(cx (o,b)), aum/av + a(x)um = 0 on r x (o,b), and b r t2m+l m 2 um(x,b) = 0 for x E R while Iu (x,t)l dt < a. Then JO u" 5 0 if and only if J f 0 where p = Iml and {An) is P (9) the set of positive eigenvalues of -l(aijvx )x + cv = hv in ij R with av/av + crv = 0 on r.

The proof is similar in part to that of Theorem 4.4. Thus, if there exists a positive eigenvalue An of the elliptic problem indicated in the statement of Theorem 4.5 such that J = P (q) 0 then set um(x,t) = tmmJ (%t)vn(x) where vn is the eigen- P function corresponding to An. Evidently u" satisfies Lmum = 0 and the required boundary conditions. On the other hand suppose

J (5b) 0 for all An as described. Let u" be a solution of P Lmum = 0 satisfying the conditions stipulated in Theorem 4.5.

Let +,(t) and vn(x) be the normalized eigenfunctions for the

(4.16) and the elliptic eigenvalue problem in the statement of

Theorem 4.5 respectively. Then the set C+,(t)v,(x)l is a complete orthonormal set for the real Hilbert space H = 2 L,(R x (o,b)) = (h; 1: 1, t2m+1h2(x,t)dxdt < m) and since urn E H we can write um(x,t) = la..+.(t)vi(x) with the standard for- 1J J mula for aij. Now define hi(t) = um(x,t)vi(x)dx so that R

260 4. SELECTED TOPICS

(t2m+’h;(t))l + Ait2m+1hi(t) = 0, hi(b) = 0, where Ai is the eigenval ue corresponding to vi. By the Parseval-Plancherel formula ]It2m’hf(t)dt = rla..I21J < m and therefore hi(t) 0 if

Ai is not an eigenvalue for the problem (4.16), i.e., if

Jp(qb) $ 0. Consequently um(x,t) z 0 since the vi(x) are com- 2 plete in L (R) and um is continuous in R x (o,b).

Remark 4.6 We remark first (cf. Travis [l]) that for m 5 -1 the problem treated in Theorem 4.5 is equivalent to Lmum = 0,

3um/3v + oum = 0 on r‘ x (o,b) and um(x,b) = um(x,o) = 0.

Remark 4.7 Let us note in passing that, in general, if L is a singular second order ordinary differential operator in t such that the numerical initial value problem Lu + Anu = 0, u(o) = 1, ut(o) = 0, has a solution un(t), when A, is an eigenvalue of the (closed densely defined) operator A in a Hilbert space E, while the corresponding set of eigenvectors {en) of A forms a complete orthonormal set in E, then formally the E valued function w(t) = 1 anun(t)en satisfies Lw + Aw = lan(Lun++,un)en = 0 with w(o) = lanen and wt(o) = 0. Choosing the an to be the Fourier coefficients of w(o) = e E D(A) one has a formal solution of Lw + Aw = 0 with w(o) = e and wt(o) = 0.

Uniqueness questions could then be handled in terms of the 2 I * L formal adjoint L of L and the operator A . Indeed given that Ae = A(lanen) = lanAen = CanAnen for e E D(A) with an =

(eyen ) we have (Ae,ek) = ~anAn(en,ek) = akAk = Ak(e,ek) so

2 61 SINGULAR AND DEGENERATE CAUCHY PROBLEMS * * - * ek E D(A ) with A ek = Akek. Hence A has a complete orthonormal set of eigenvectors ek with eigenvalues Tk. Assume then I that the problem L v + Tnv = 0, v(b) = 0, v (b) = 1 has a solu- t I1 I tion vn(t) on [o,b] for any b. Here, given Lu = u + au + I 2 Bu with u(o) = u (0) = 0 and v a C function satisfying v(b) = I I II I 0 and v (b) = 0 we define L v = v - (w) + Bv and observe b ti that i u v dt = -u(b) + On Now write w = lbnvn(t)en where e = w (b) = lbnen is an arbitrary * element of some dense set (e.g., D(A )). Then if (L+A)w = 0 I with w(o) = w (0) = 0 we consider formally the equation

b A b (4.18) 0 = ((L+A)w,w)dt = lbn I (Lw,vnen)dt 0 0

bl + 1 (w,L vnen)dtl + lb 0

It follows that w(b) = 0 for any b and hence w z 0. The calcu-

lations have been formal but they apply to realistic situations.

m Theorem 4.8 Assume m E (-l,l),Lmu = 0, urn E C2(sZx(o,b)),

aum/av + a(x)um = 0 on r x (o,b),um(x,b) = 0, and

lim t2m+1W(um,~,t) = 0 as t + 0 where $ is a possibly complex

262 4. SELECTEDTOPICS valued solution of the differential equation in (4.17) for some

A = A satisfying lim t2m+1W(q,~,t) = 0 as t + 0. Then u" :0 0 if and only if Rn f Ak where Rn and Ak are the eigenvalues of

(4.17) and the elliptic problem in the statement of Theorem 4.5 respect ivel y . The proof here is similar to that of Theorem 4.5. Finally

Travis [l]states some results related tp Theorems 4.3 and 4.4 and some comparison is instructive (see Remark 4.10).

Theorem 4.9 Let u E C2(Ex(o,b)) satisfy Lmum = 0 and

3um/3v + a(x)um = 0 on r x (o,b), while um(x,o) = um(x,b) = 0 for x E R when m < 0 or Ium(x,o)l < m and um(x,b) = 0 for x E R when m 2 0. Then urn E 0 if and only if J f 0 where the P (q) An are the eigenvalues of the elliptic problem in the statement of Theorem 4.5 and p = Im

Remark 4.10 Theorem 4.9 (rephrased for urn = 0 on r x [o,b)) is independent of Theorem 4.3 for example even when - 1/2 < m < m 0. In Theorem 4.3 one assumes ut is continuous in E x [o,b] whereas Theorem 4.9 assumes only continuity of u: on Ex (o,b).

Consider for example um(x,t) = t'mJ(t) sin x for 1/2 < m < 0 P - m 2m+l m m m which satisfies utt + T~- uxx = 0 with u (o,t) = u"(r,t) =

0 and um (x,o) = 0. But uT(x,t) = O(t-2m-1) as t + 0 so the hypotheses of Theorem 4.3 do not hold whereas Theorem 4.9 (rephrased) will apply. On the other hand Theorem 4.3 does not require conditions at t = b for m > -1. Similarly Travis [l] 2 states a result for a Neumann problem (ut(x,o) = ut(x.b) = 0)

263 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

related to a theorem of Young [4].

4.5 Miscellaneous problems. Various Cauchy problems with boundary data for EPD equations have been studied and we will mention a few results. First consider (cf. Fusaro [41)

Problem 5.1 Solve Lmum = utt + ___2m + urn um = 0, where t t - xx 2 m > - 1/2, o < x < r, t > 0, um(x,o) = f(x) (f E c 1, u! (x,o) =

0, and um(o,t) = um(r,t) = 0. One assumes also f(o) = f(n) = 0.

Then by separation of variables Fusaro [41 obtains a

solution in the form

m -m (5.1) um(x,t) = r(m+l) 1 bn [F] Jm(nt) sinnx n=l IT where bn = [:] I f(x) sinnx dx. Using the formula (1.3.7) with 0 a change of variables $ = r/2 - 0 and putting this in (5.1) one obtains

2m cos $ d $

where 8 denotes the Beta function. There results (cf. Fusaro

[41)

Theorem 5.2 A unique solution of Problem 5.1 is given by

(5.2).

The proof of Theorem 5.2 is straightforward and will be 1 omitted. If one sets 5 = sin $ and M(x,a,f) = 2 [f(x+a) + f(x-o)] we can write (5.2) in the form

264 4. SELECTEDTOPICS

(5.3) um(x,t) = 2~[+, m + ~]-1~1(l-~)m-"2M(x,Et,f)dg 0 which coincides with (1.6.3). We remark here that f is extended from (O,IT) as an odd periodic function of period ZIT. Another type of such "mixed" problems is solved by Young

191 ; here mixed problem means mixed initial-boundary value prob-

lem and does not mean that the problem changes type. Thus consider

m 2m+1 = Problem 5.3 Solve Lmum = utt + ___ - t t xx YY f(x,y,t) in the quarter space x > 0, t > 0, - 03 < y < m, where m m > - 1/2, u (x,y,o) = u:(x,y,o) = 0, and u"(o,y,t) = g(y,t) with 2 f E C2, g E C , and g(y,o) = gt(y,o) = 0.

At points (x,y,t) where x 2 t > o the presence of g is not felt and Young takes the solution in the form given by Diaz-

Ludford [31

where D is the domain bounded by T = Oandthe retrograde characteristic cone R = (t-T)' - (x-c)' - (y-0)' = 0 (t-oO) with vertex at (x,y,t) while ^R = (tt-r)' - (x-c)' - (y-n) 2 . In the region where 0 < x < t formula (5.4) no longer gives the solution since it does not involve g and Young uses a method of M. Riesz as adapted by Young [l;21 and Davis [l;21 to adjoin additional

terms to the um of (5.4). We will not go into the derivation of the final formula but simply state the result; for suitable m

265 SINGULAR AND DEGENERATE CAUCHY PROBLEMS the technique can be extended to more space variables. The solution is given by the formula

22m+l a --_ 'lr ax IT 2 where R, = (t-.)' - (x+<) - (y-rl)', Ro (resp. Go) denotes the I\ 2 value of R (resp. R) for 5 = 0, i* = (ttT)2 - (x+E)2 - (y-rl) y

D'. is the domain where < > 0, T > 0, bounded by the retrograde * cone with vertex at (-x,y,t)(thus D C D), and T is the region

intercepted on the plane 5 = 0 by the cone R = 0 (T > 0). Note * that for x 2 t > 0 D and T are empty while as x + 0 the inte- * * h grals over D and D cancel each other since D + D and (R,R) +

Theorem 5.4 The solution of Problem 5.3 is given by (5.5).

It is interesting to note what happens when all of the data

in an EPD type problem are analytic. Thus for example Fusaro

[3] considers for t > 0, 1 5 k 5 n, and x = (xl, . . ., Xn)

(5.6) u"tt + 2m+lt umt = F(x,t,um,Dku",u~,Dku~,DkDeum);

m m u (x,o) = f(x); Ut(X,O) = 0

266 4. SELECTEDTOPICS where Dk = a/axk while F and f are analytic in all arguments.

Theorem 5.5 The problem (5.6) has a unique analytic solution for m -1, -3/2, -2, . . . .

Corollary 5.6 If F(x,t,u, . . .) = G(x,u,Dku,D kaD u) + g(x,t) + h(t)ut with g even in t and h odd in t, then the problem (5.6) has for all m f -1, -3/2, -2, . . . a unique analytic solution and it is even in t. If m = -3/2,-5/2, . . . there exists a unique solution in the class of analytic functions which are even in t.

Fusaro [3] uses a Cauchy-Kowalewski technique to prove these results and we refer to his paper for details. One should note also the following results of Lions [l],obtained by transmutation, which illustrate further the role played by even and Cm functions of t in the uniqueness question for EPD equations.

Let E, be the space of Cm functions f on [o,m) such that f(2n+1)(o) = 0 (thus f E E, admits an even Cm extension to

(-0,m)) and give E, the topology induced by E on (-,a). Let H be a Hilbert space and A a self adjoint operator in H such that for all h c D(A) (Ah,h) + allh112 Z 0 for some real a. Setting

Lm=D D and putting the graph topology on D(A) Lions +-*'+' t proves

Theorem 5.7 For any m E f, m =f -1, -2, -3, . . ., there h exists a unique solution um E E, BT D(A) of Lmum + hum = 0 for

267 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

t 2 0 with um(o) = h E D(A") and uT(o) = 0.

Here T denotes the projective limit topology of Grothendieck

(cf. Schwartz [5]) which coincides with the E topology since E, is nuclear. If m f -1, -3/2, -2, . . . and E+ is the space of Coo functions on [o,m) with the standard E-type topology then any

A solution urn E E+B~ D(A) of L~U"' + hum = o satisfying um(o) =

A h E D(A") and uT(o) = 0 is automatically in E*BT D(A) so the * corresponding problem in E+ BT D(A) has a unique solution. For the exceptional values m = -1, -2, -3, . . . Lions [l]proves

Theorem 5.8 Let m = -(p+l) with p 2 0 an integer so that

h 2m + 1 -(2p+l). There exists a unique solution urn E E+qD(A) m m 2p+l m of L,u + Au = 0, um(o) = uY(0) = . . . = Ut u (0) = 0,

DZP+*um(o) = h E D(Am), and DZP+3um(o) = 0.

In connection with the uniqueness question we mention also the following recent result of Donaldson-Goldstein [Z]. Let H be a Hilbert space and S a self adjoint operator in H with no nontrivial null space. Let Lm = D2 + D again and define urn 2m+lt to be a type A solution of

m m (5.7) LmU + S2um = 0; um(o) = h E D(S2); ut(o) = 0

2 1 if urn E C ([o,m),H) and a type B solution if urn E C ([o,m),H)n

C2( (oYrn),HI.

Theorem 5.9 The problem (5.7) can have at most one type' A solution for m 2 -1.

268 4. SELECTED TOPICS

t-2mu-m Further, by virtue of (1.4.10) and the formula um =

(cf. Remark 1.4.7), for m < -1/2 not a negative integer there is no uniqueness for type B solutions. We note here that for -1 < m < -1/2 we have 1 < -2m < 2 and um = t -2mu-m = O(t-2m) -+ 0 as

t + 0 with u" = O(t-2m-1 ) -+ 0 but uYt = O(t-2m-2) -+ m so such a t m 21 u is not of type A. For m = -1 we look at u-l = t u with

-1 -1 O u (0) = u;' (0) = 0 and utt E C ([o,m) ,H). However from Remark

1.4.8 we see that the solution u-' arising from the resolvant

R-l (which could be spectralized as in Section 1.5) does not have a continuous second derivative in t unless the logarithmic term 2 vanishes, which corresponds to S h = 0 in the present situation.

Thus in general (5.7) does not seem to have a type A solution for 2 m = -1 and if it does the solution arises when S h = 0. Indeed if L-lu-l + S2u-l = 0 with u-l(o) = h and u;l(o) is well defined then U;;(O) = lim l/t u,'(t) as t + 0 so L lu-l = u-l u-l = tt - 1t t 2 -1 2 -S2um1 -+ 0 which means that S u (0) = S h = 0 (cf. Weinstein

[3]), which is precluded unless h = 0. Note that if h 0 with 2- 2 S h - 0 then u-'(t) = h and u-'(t) = h + t h will both be solutions of (5.7).

We go now to some work of Bragg [2; 3; 4; 51 on index shifting relations for singular problems. This is connected to the idea of "related" differential equations developed by Bragg and

Dettman, loc. cit., and described briefly in Chapter 1 (cf. Section 1.7). Consider first for example (a 2 0, u 2 1)

269 SINGULAR AND DEGENERATE CAUCHY PROBEMS

This is a radially symmetric form of the EPD equation when = n for example since the radial Laplacian in Rn has the form

AR = 2 + n-1 As Dr -r Dr* initial data we take

(5.9) U (ay’)(ryo) = T(r); u(ay’)(ryo)t = 0

From the recursion relations of Chapter 1 (cf. (1.3.11), (1.3.12), (1.3.54), (1.3.55), (1.4.6), (1.4.10), and Remark 1.4.7) we see that there are certain “automatic” shifts in the a index.

For example the Sonine formula (1.4.6) with al = 2p + 1 and al + a2 = 2m + 1 yields for al 2 0, a2 > 0 a2-2 (a1+a2 ,!J) -al -a2 t2-,,2) 2;1 (5.10) u (r,t) = c1,2t It( 0

(a1 YU) ‘U (rYrl) drl

al+l a2 where c = 2/f3(~ Bragg [5] obtains this result using 12 , 7). Laplace transforms and a related radial heat equation

(5.11) v: = vyr + $!-v:; v’(r,o) = T(r)

The connection formula for a 2 1 is given under suitable hypotheses by (cf. Bragg [5], Bragg-Dettman [6])

2 (here L;’ is the inverse Laplace transform with kernel est ) arid (5.10) follows directly by a convolution argument. Similarly, setting Iv(z) = exp, (-Fv~i)1 Jv(iz) and Ku(r,S,t) =

270 4. SELECTEDTOPICS

22 (a)-lrl-’/2<’/2 exP C-(r +E )/4tlIu/2-1 (rE/2t) , one can show P that v’(r,t) = r,E,t)T(E)d< satisfies (5.11), and using 1 oKJ (5.12) there results from known properties of Iv

Theorem 5.10 Let a 2 0 with u(~’’) a continuous solution of (5.8) - (5.9). Then

satisfies (5.8) = (5.9) with index 1-1 + 2. On the other hand a direct substitution shows that if u (ay4-’)(r,t) satisfies (5.8) with index 4 - u and u (ay4-’)(r,0) = r’-2T(r) then u(ay’)(r,t) = r 2-’u(ay4-’)(r,t) satisfies (5.8) - (5.9) except possibly at r = 0.

Using the index shifting relations indicated on a and p Bragg [5] proves a uniqueness theorem for (5.8) - (5.9) when a 2 0 and 1-1 L 1 and constructs fundamental solutions. The idea of related differential problems such as (5.8) - (5.9) and (5.11) (or (5.8) - (5.9) with index changes), having connnection formulas like (5.12) (or (5.10)), has been exploited by Bragg [2;

3; 41 to produce a variety of index shifting relations involving at times the generalized hypergeornetri c functions

where (a), = a(a+l) . . . (a+n-1), II:(yj),, = 1 if p = 0, and none of the 6. are nonposi tive integers (cf. Appell -Kampe/ de J

271 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Fgriet [l]).We will mention a few results in this direction without giving proofs and here we recall that if 0 = zDZ then w = F satisfies the differential equation Pq IOnl(O+B.-l)-zn~(O+a.)}wq = 0. Thus let P(x,D) = J A: ‘n raX(x)D1 . . . Dn where Dk = a/axk and [XI = lXi 5 m. Let el(qYB,t,Dt) = tD llq(tD +B.-l) and 02(pya,t,Dt) = tIIy(tD +a ). t1 tJ tj Then given P(x,D) and Q(x,D) of the type indicated with orders

R1 and R2 respectively and r = max (p+R1 ,q+R2) with p 5 q one can prove for example (cf. Bragg [4])

Theorem 5.11 Let u E Cr in (x,t) for t > 0 with u and all its derivatives through order r bounded and lim u(x,t) = T(x) as t + 0. Suppose Q(x,D)01(q,f3,t,Dt)u - P(x,D)0,(p-1,a,t,Dt)u = 0. -1 rrn -0 -1 Then for a > 0 the function v(x,t) = r(a ) e u P u(x,tu)du P p 10 satisfies Qel(q,B,t,Dt)v - P82(p,art,Dt)v = 0 and v(x,o) = T(x). Similarly, under suitable hypotheses, if [Q0 (4-1 ,B,t,Dt) - 1-Bq q P02(p,a,t,Dt)]u = 0 and w(x,t) = t r(Bq)Ui1[S -A u(~,l/s)]~+~ then one has [Qel(q,B,tyDt) - P02(p,a,t,Dt)]w = 0. Index shift formulas in (p,q), a, and are also proved and when applied to

EPD equations for example will yield some of the standard recursion relations. We note that under the change of variables 12 a 5 = t the EPD equation utt + ut = Pu becomes [

Bragg [3] where a more systematic theory is developed; nonhomogeneous problems are studied in Bragg [2].

272 4. SELECTEDTOPICS

We conclude with a few remarks about nonlinear EPD equations.

In Keller [l]it is shown that for m > 0 and certain nonlinear f the Cauchy problem

2 (5.15) L u = utt u c Au = f(u); m + -"+'tt -

u(x,o) = uo(x); Ut(X;O) = 0 does not have global solutions in R2 x (0,m) for arbitrary ini- 1 tial data uo. Levine [2] studies this problem for - < m 5 0 and x E R c Rn with u = 0 on r x [o,T) where R is bounded with 1 sufficiently smooth boundary r. Let f be a real valued C function and for each real @ E C2(E) define G(@) = 1 (~(x)f(r)dz)dx. r RO Set (@,I)) = 1 @(x)$(x)dx and assume there is a constant ct > 0 R such that for all @ E C2(E)

It can be shown that if, for some ct > 0 and some monotone increasing function h, f(z) = l~1~~+'h(z),then (5.1G) is valid.

Assuming the existence of a local solution for each uo E C2(E) vanishing on r Levine proves

1 Theorem 5.12 Let - 2 < m 5 0 and assume (5.16) holds. If - u : R x [o,T) -+ R1 is a classical solution of Lmu = f(u) with u(x,o) = u (x), ut(x,o) = 0, and u = 0 on x [o,T), while 0 r 1 2 G(uo) > 7 ~RIVuoldx, then necessarily T < as t -+ T-.

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327 This page intentionally left blank Index

Accretive, 180, 219, 220, 222 A adapted, 70 Adjoint method, 70 Almost subharmonic function, 36, 38 Analytic data, 266 Analytic semigroup, 180 Antidual, 179, 187 A priori estimate, 190 Associate resol van t 55 Backward Cauchy pro lem, 173 Backward parabolic, 203 Baire function, 150 Basic measure, 150 Basis elements, 99, 109, 115, 126, 138 Biinvariance, 94 Boundary, 91 , 106, 25 Boundary condition of third type , 226 Boundary condition of fourth type, 226 Bounded, 207 Calculus of variations operator, 210 Canonical recursion relations, 97, 104, 129 Canonical sequence, 97, 101, 121, 129 Canonical triple, 105, 112 Carrier space, 149 Cartan decomposition, 91, 105, 121, 124 Cartan involution, 105, 108 Cartan subalgebra, 106, 112, 121 Casimir operator, 97, 117, 121 Cauchy integral formula, 24 Cauchy Kowalewski theorem, 267 Center, 121 Centralizer, 91, 106, 124 Character, 93 Characteristic, 200 Characteristic conoid, 238, 244, 265 Coercive, 181, 202, 207 Compatible operator, 212, 213 Connection formulas, 29 Contiguity relations, 133 Composition, 11 9 Darboux equation, 93, 119, 120, 140 Degenerate problem, iii Demi mono tone , 21 0 Diagonalizable operator, 47, 150

329 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Differential inclusion, 219, 229 Diffusion, 143, 145 Directional oscillations, 162 Dirichlet condition, 226 Dirichlet problem, 255 Distinguished boundary, 24 Doubly degenerate, 228 Doubly nonlinear equations, 177, 229 Eigenvalue equation, 97, 101, 253, 257, 259, 261 Eisenstein integral, 99 Elliptic parabolic problem, 199, 226 Elliptic parabolic Sobolev problem, 201 Elliptic region, 5 Energy methods, 43, 164 Entrance boundary, 49 Enveloping algebra, 97, 118, 121 Euclidean case, 136 Euler angles, 112 Euler Poisson Darboux (EPD) equation, 1, 4, 9, 67 Exceptional cases, 34 Exponential type, 29 Finite part (Pf), 249 Flow, 143, 146 Fourier transform, 9, 92 Fractional powers, 186 Fredholm a1 ternative, 259 Fundamental matrix, 14 Gas dynamics, 5 GASPT, 8 Gelfand map, 149 Generalized hypergeometric function, 271 Generalized radiation problem of Weinstein, 245 Genetics, 145, 204 Geodesic, 107, 116, 19, 137, 140 Gravity waves, 145 Green's matrix, 151 Gronwall ' s 1 emma, 53 Growth and convexity theorems, 35, 42, 78, 139 Harish Chandra funct on, 92, 108 Harmonic space, 119, 140 Hartog' s theorem, 11 Hemicontinuous. 175. 177, 207, 223 Huygen s'operator , 240 Huygens' principle, 237 Huygens' type operator, 238 Hyperaccretive, 231 Hyperbol ic region, 4 Hypergeometric equation, 134, 242 Hypocontinuous, 10, 13, 28, 30, 63, 159 Hypoelliptic, 9

330 INDEX

Improperly posed problems, 251 Index shifting, 269 Induced representation, 113, 137 Integrating factor, 16, 49 Interpolation spaces, 186 Invariant differential operators, 90, 94, 122 Invariant integrat ion, 94 Isotropy group, 93, 107 Iwasawa decomposition, 91 , 106, 124 Joint spectrum, 149 Killing form, 92, 105, 122 Kortweg deVries equation , 184 Kummer relation, 101, 129, 131 Laplace operator, 1, 97, 117, 120, 121, 138, 270 Laplace Beltrami operator, 117, 122, 140 Laplace transform, 270 Lauricella function, 243 Light cone, 122, 238 Limit circle, 258 Limit point, 258 Lions' projection theorem, 45, 165, 191 Lipschitz condition, 167, 170 Lobacevs kij space, 122 , 128 Local ly equi continuous semi group, 66, 79 Long waves, 171 Lorentz distance, 243 Lorentz group, 122 Maximal abelian subspace, 91 , 106, 124 Maximal compact subgroup, 91, 122 Maximal ideal, 149 Maximal monotone , 208 Maximum principle, 42 Mean value, 5, 8, 93, 97, 118, 119, 139, 237 Metaharmonic, 36 Metaparabol ic , 184 Measurable family of Hilbert spaces, 47, 150 Measurable family of projections, 177 Metric tensor, 107, 140 Mixed problems, 5 Monotone, 177, 180, 206, 220 Multivalued function, 217 Multivalued operator, 216, 230 Negative curvature, 119, 140 Neumann condition, 226 Neumann problem, 258, 263 Neumann series, 16, 20, 49, 56, 239 Nonlinear EPD equations, 65, 273 Nonlinear semigroup, 216 Normalizer, 91, 106, 124 Operator of type My 213, 215

331 SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Orbit, 93 Osgood function, 66 Paley Wiener Schwartz theorem, 30 Parabolic line, 4, 147 Parabolic subgroup, 93 Parseval Plancherel formula, 261 Periodic condition, 209, 216 Pizzetti Zalcman formula, 96 Point spectrum 252 Polar coordinates, 91, 95, 99, 113, 116, 137, 240 Pol ar deoomposi tion , 105 Positive measure, 36 Principal series, 93, 114 Protter condition, 161 Pseudomonotone, 209, 212 Pseudo parabol i c , 143 Pseudosphere, 122 Quasiregular representation, 93 Radial component, 95, 96, 97, 99, 100, 110, 246 Radial inertia, 175 Radiation, 245 Rank, 91 Recursion formulas, 12, 29, 33, 34, 70, 120, 129, 130, 132, 272 Regular family of sesquilinear forms, 188, 204 Regul arize, 172 Regularly accretive, 186 Related differential equations, 2, 81, 269 Relation, 217 Resolvant, 10, 14, 29, 31, 48, 55, 67, 71, 96, 101, 111, 115, 128, 130, 132, 135, 151, 239, 241 Resolvant initial conditions, 10, 17, 97, 110, 128, 132, 139 Riernann Liouville integral , 74, 272 Riemannian structure, 97, 107, 118, 137 Riesz map, 174, 179, 180, 196, 217 Riesz operators, 162 Right angle condition, 231 Root spaces, 91, 106, 122, 124 Rotating vessel , 174 Scalar derivative , 7 Section of a distribution, 37 Sectorial operator , 180 Semi di rect product, 136 Semi 1 inear degenerate equations, 206 , 228 Semi 1 i near singular problems, 65 Semimonotone, 209 Semisirnple Lie group, 91 Separately continuous, 7, 63, 159 Sequential completeness, 71 Simple convergence, 61 Simply continuous , 61

332 INDEX

Singular elliptic equation, 5 Singular problem, iii Sobolev equation, iv, 143, 144, 174, 200, 227 Sobolev space, 224 Sonine integral formula, 32, 34, 76, 117, 119, 270 Spectral decomposition, 47, 150 Spectral techniques, 47, 147, 162 Splitting relations, 101, 104, 130, 132 Stationary problem, 214 Strictly monotone, 209, 214 Strong derivative, 7 Strongly continuous, 61, 168 Strongly monotone, 176, 229 Strongly regular, 147, 166 Subdifferential, 232, 234, 235 Subgradient, 232 Symmetric space, 91 Test function, 5 Topology E, 69 Topology T, 268 Transmutation, 2, 80, 248 Transverse derivative, 260 Tricomi equation, 4 Type A solution, 268 Type B solution, 268 Uniformly well posed, 28 Unimodular, 94 Unimodular quasiunitary matrices, 112 Value of a distribution, 37 Variable domains, 175, 177, 192, 194 Variational boundary conditions, 196 Variational inequality, 175, 178, 209, 211 V elliptfc, 182 Vibrations, 175 Viscoelasticity, 185, 227 Von Neumann algebra, 150 Wave equation, 1, 42, 146 Weak problem, 44, 47, 161, 164 Weakly continuous operator, 213 Weakly regular, 147, 179 Weinstein complex, 77 Weyl chamber, 91 Weyl group, 92, 106, 125 Winter function, 66 Yosida approximation, 173 Zonal spherical function, 94 A6 07 C8 D9 EO F1 62 H3 14 15 333 This page intentionally left blank