Journal of Mathematical Analysis and Applications 255, 678᎐697Ž. 2001 doi:10.1006rjmaa.2000.7308, available online at http:rrwww.idealibrary.com on
A Singular Perturbation Problem of Carrier and Pearson
Walter G. Kelley View metadata, citation and similar papers at core.ac.uk brought to you by CORE
Uni¨ersity of Oklahoma, Norman, Oklahoma 73019 provided by Elsevier - Publisher Connector
Submitted by Robert O’Malley
Received October 5, 1999
1. INTRODUCTION
The use of perturbation methods to study singularly perturbed differen- tial equations has enjoyed spectacular success since its origination in the work of Prandtlwx 19 on viscous incompressible flow past an object for a large Reynolds number. During the nearly 100-year history of this area, a large number of formal approximation methods have been developed and refined, including the method of averaging, boundary layer methods, matched asymptotic expansions, and multiple scales. The textbooks of Nayfehwx 15 , O’Malley wx 18 , Kevorkian and Cole wx 11 , Bender and Orszag wx1 , Smith w 20 x , and Holmeswx 5 contain discussions of some or all of these methods with many examples that illustrate the fascinating variety of behavior exhibited by these problems. On the other hand, examples are known for which the formal methods fail, either because the expansions contain terms that cannot be deter- mined uniquelyŽ. so that the method leads to ambiguous results or because the computed expansions are spuriousŽ i.e., do not approximate actual solutions of the problem. . One of the most famous examples of the failure of the formal methods is a problem introduced in a textbook of Carrier and Pearsonwx 3 ,
⑀ 2 uЉ q u2 y 1 s 0, uŽ.y1 su Ž.1 s 0.
678 0022-247Xr01 $35.00 Copyright ᮊ 2001 by Academic Press All rights of reproduction in any form reserved. A SINGULAR PERTURBATION PROBLEM 679
They constructed an approximate solution q y 1 xxx0 AxŽ., ⑀ sy1 q 3 sech22q lnŽ.'' 2 q 3 q 3 sech ''2 ⑀ 2 ⑀ 1 y x q 3 sech2 q lnŽ.'' 2 q 3, '2 ⑀
y where x0 is an arbitrary number in Ž.1, 1 . This approximation is near y s " s 1, except for boundary layers at x 1 and a spike layer at x x0. One can show that AxŽ., ⑀ satisfies the boundary value problem except for exponentially small remainder terms. However, since the exact solutions of the problem can be expressed in terms of elliptic functions, which are s periodic, only the choice x0 0 corresponds to an actual solution. In fact, O’Malleywx 17 used phase plane analysis to show that any spike layers for autonomous problems of this type must occur at evenly-spaced points in the interval. The Carrier᎐Pearson example has been discussed in many papers. Langewx 12 used a formal matching procedure that was special to the problem under consideration and included exponentially small terms to resolve the ambiguity. Variational methodsŽ see Kath et al.wx 8. lead to a similar conclusion but do not provide detailed approximations. Wardw 21, 22x demonstrated that the indeterminacy can be resolved for this and related problems by an asymptotic projection method. Most recently, MacGillivraywx 13 presented a systematic matching scheme involving expo- nentially small terms that determined the position of the spike as a product of the matching process. In addition, there is a large body of work on the existence and location of spikes for positive solutions of semilinear elliptic boundary value prob- lems. For the case where the nonlinearity has three zeros, Kelley and Ko wx9 and Jang wx 6 provided a number of existence results for the Dirichlet problem. Ni and Weiwx 16 and Wei wx 23Ž see also the many other works cited in these papers. used variational methods to investigate the existence and location of single or multiple spikes for certain Dirichlet problems in which the nonlinearity has a pair of zeros. The Neumann problem has received even more attention because of its importance as a mathematical model for morphogenesisŽ see Gui and Weiwx 4 and the many references therein. . In this paper, we reconsider the Carrier᎐Pearson example from a different point of view. Working with a general autonomous problem, we show that the assumption of a symmetric location of the spike allows the construction of uniform approximate solutions to all exponentially small orders. Moreover, the existence of these formal approximations permits 680 WALTER G. KELLEY their rigorous validation by means of a simple Green’s function argument. One of the interesting details of the argument is that the Green’s function is exponentially large, from which it follows that approximate solutions with exponentially small errors may be spurious if there is no procedure available for improving the approximation. Our analysis does not establish the nonexistence of solutions with spikes located elsewhere, but it will be clear that the assumption of symmetry is essential to our construction of higher order approximations. The final section of this paper contains some related examples. The first two deal with the issues of nonsymmetric boundary conditions and multi- ple spikes. The last one is an example from Carrierwx 2 involving a nonautonomous differential equation. This example has also been dis- cussed by Kathwx 7 , Bender and Orszag wx 1 , and most recently MacGillivray et al.wx 14 . Once again formal methods produce approximate solutions with spikes at arbitrary interior points. However, in this case the problem has solutions with spikes only at the endpoints and near the midpoint of the interval.
2. CONSTRUCTION OF AN APPROXIMATION
We will construct an approximate solution with a spike near x s 0 for the following boundary value problem with a small, positive parameter ⑀ multiplying the highest derivative: ⑀ 2 uЉ q fuŽ., ⑀ s 0,Ž. 2.1 uŽ.y1 su Ž.1 s 0.Ž. 2.2 Our basic assumptions are that fuŽ.Ž, ⑀ s 1 q g Ž..Ž.⑀ Fu, where g Ž.⑀ ª 0 as ⑀ ª 0, and that F satisfies the following hypotheses. There are numbers a - 0 - b such that Ž.H1 Fa Ž.s 0, Fb Ž.) 0; Ž.H2 FЈ Ž.a - 0; u - - - b s Ž.H3 HaaFsdsŽ. 0 for a u b and H FsdsŽ. 0; Ž.H4 F has derivatives of all orders. It is well known thatŽ. H3 is a necessary condition for the existence of a solution with a spike extending from near u s a up to u s b Žsee O’Mal- leywx 17. . The following elementary lemma will be useful in the construction.
LEMMA 2.1. Let hŽ. x and j Ž. x be continuous and e¨en on Žyϱ, ϱ .. Then any solution yŽ. x of yЉ q hxy Ž.s j Ž. x with yЈ Ž.0 s 0 is also e¨en. A SINGULAR PERTURBATION PROBLEM 681
The construction of the initial approximation is fairly standard, so we give only a sketch. First, the approximation a q SxŽ.r⑀, ⑀ near the spike at x s 0 is given implicitly by << aqS d¨ x sy . H y ¨ ⑀ b ' 2Ha f
One can show that a q SxŽ.r⑀, ⑀ is a smooth solution of Eq.Ž. 2.1 and even, and that
yl Ž⑀ .< x < r⑀ F r⑀ ⑀ F yl Ž⑀ .< x < r⑀ Ce12SxŽ., Ce y F F ⑀ syЈ ⑀ for 1 x 1, where lŽ. ' f Ž.a, and C12, C are positive con- stants. Similarly, the boundary layer approximation a q LŽŽ1 q x .r⑀, ⑀ . at x sy1 is given implicitly by
q d¨ 1 q x a L s H yy ¨ ⑀ 0 ' 2Ha f for y1 F x F 1if a - 0. If a ) 0, we simply change the sign on one side of the equation. Then a q L is a smooth solution of Eq.Ž. 2.1 , satisfies the left boundary condition, and
yl Ž⑀ .Ž1qx.r⑀ F q r⑀ ⑀ F yl Ž⑀ .Ž1qx.r⑀ De12LŽ.Ž.1 x , De y F F for 1 x 1 and some positive constants D12, D . By symmetry, the approximation at x s 1is a q RŽ.Ž.Ž.1 y x r⑀ , ⑀ ' a q L Ž.1 y x r⑀ , ⑀ . Now the initial approximation is ⑀ s q r⑀ ⑀ q q r⑀ ⑀ q y r⑀ ⑀ Ax0 Ž., a Sx Ž, .LŽ.Ž. Ž.1 x , R Ž.1 x , , Ž.2.3 an even function.
To determine the accuracy of A0 , consider ⑀ 2 Y q ⑀ s ⑀ 2 Љ q Љ q Љ q q q q ⑀ A00fAŽ.Ž, S L R .Žfa S L R, .. On the interval wxy1, 0 , we have ⑀ 2 RЉ, R s OŽeyl Ž⑀ .r⑀ .,as⑀ ª 0, so
⑀ 2 Y q ⑀ s q q ⑀ y q ⑀ y q ⑀ A00fAŽ.Ž, fa S L, .Žfa S, .Žfa L, . q O Ž.eyl r⑀ . 682 WALTER G. KELLEY
The mean value theorem, together with the properties of S and L, implies that the expression in brackets is also OŽeyl r⑀ .. A similar argument on the intervalwx 0, 1 then yields
⑀ 2 Y q ⑀ s yl r⑀ A00fAŽ.Ž., O e wxy ⑀ ª uniformly on 1, 1 as 0. A0 also satisfies the boundary conditions except for exponentially small terms. If S0 were not centered at 0 then the error term would be larger, but still exponentially small. Next, we seek better approximations of the form
N ⑀ s ⑀ q r⑀ ⑀ q q r⑀ ⑀ AxN Ž., Ax0 Ž., Ý Txkk Ž, .U Ž. Ž1 x ., ks1 eyklr⑀ qV 1 y x r⑀ , ⑀ .2.4 k Ž.Ž. ⑀ ky1 Ž.
ŽSince all of our functions depend on ⑀, we will usually omit ⑀ in our functional notation.. We will show how the first few terms in the approxi- mation are computed, and it should be evident that higher order terms can be computed in a similar way. SubstitutingŽ. 2.4 into Ž. 2.1 , we have
⑀ 2 Y q s ⑀ 2 Y q ⑀ 2 Y q Y q Y yl r⑀ ANNfAŽ.A0111 ŽT U V .e N eyklr⑀ q ⑀ 2 T q U q V Љ q fA, ⑀ Ý Ž.Ž.kkk⑀ ky1 0 ks2 N eyklr⑀ q fЈ A , ⑀ T q U q V q иии . Ž.Ž0 Ý kkk .⑀ ky1 ks1
⑀ 2 Y q We will define T100so that A fAŽ.is eliminated from the preced- ing expression. Let s xr⑀, and let a dot indicate differentiation with respect to . Then we want
q Ј q sy ⑀ 2 Y q lr⑀ T¨1100f Ž.a ST Ž.A fAŽ. e ,2.5 Ž. s T˙1Ž.0 0.Ž. 2.6
Now S˙ is a solution of the homogeneous equation corresponding to Eq. Ž.2.5 , so a solution of Eq. Ž. 2.5 is defined as
dt t ⑀ s y ⑀ 2 Y q lr⑀ pŽ., S˙˙ Ž. Ž.A00fAŽ. e Swdw Ž., HHy 2 y1 1 St˙ Ž. ⑀ A SINGULAR PERTURBATION PROBLEM 683 for - 0, and as
dt t ⑀ s ˙˙ y ⑀ 2 Y q lr⑀ pŽ., S Ž.HHŽ.A00fAŽ. e Swdw Ž., 2 1 1 St˙ Ž. ⑀ for ) 0. Since the integrand of the inner integral in the definition of p is odd, we have, by l’Hospital’s rule,
1 0 s ⑀ 2 Y q lr⑀ lim pŽ. Ž.A00fAŽ.Ž. e Swdw˙ Hy1 0 S¨Ž.0 ⑀
1 0 s ⑀ 2 Y q lr⑀ ˙ H Ž.A00fAŽ. e Swdw Ž. 1 S¨Ž.0 ⑀ s lim pŽ. , x0 so p is continuous at s 0 and solvesŽ. 2.5 for / 0. Ž Note that the symmetry of A0 is essential here.. Also, one can verify that p is uniformly bounded in ⑀ and for y1r⑀ F F 1r⑀ and sufficiently small values of ⑀ and that p has right- and left-hand derivatives at s 0. Now define, for ) 0, y˙pq Ž.0 T Ž. s S˙Ž. q p Ž. , 1 S¨Ž.0 - with a similar definition for 0, so that T1 satisfies Eq.Ž. 2.6 , is uniformly bounded, and is even by Lemma 2.1.
Now AN satisfies
⑀ 2 Y q ANNfAŽ.
N yklr⑀ YY e s ⑀ 2 U q Veylr⑀ q ⑀ 2 T q U q V Љ Ž.11 Ý Žkkk .⑀ ky1 ks2 N eyklr⑀ q fЈ ATq U q V y fЈ a q STeyl r⑀ Ž.0 Ý Žkkk .⑀ ky1 Ž .1 ks1
2 1 N eyklr⑀ q fЉ Ž.AT Žq U q V . q иии . 2 0 Ý kkk⑀ ky1 ž/ks1 By the Mean Value Theorem, Ј y Ј q s Љ q f Ž.A0 f Ža S .f Ž.cx Ž.Ž L R . 684 WALTER G. KELLEY for some cxŽ., where fЉ ŽŽ..cx is even, and we have at this stage ⑀ 2 Y q s q yl r⑀ ANNfAŽ.O Ž. ŽL Re . . In order to improve the accuracy and left-hand boundary value of the s q r⑀ approximation, choose U1Ž., with Žx 1 . and a dot indicating differentiation with respect to , to satisfy q Ј q sy Љ r⑀ U¨111f Ž.Ž.LŽ. aU f cxŽ. T Ž x .L Ž., sy y r⑀ q r⑀ lr⑀ y y r⑀ U11Ž.Ž.Ž.0 Ž.S 1 R 2 e T Ž.1 . Specifically, we define elr⑀ Ž.RŽ.Ž.Ž.2r⑀ q S y1r⑀ q T y1r⑀ U Ž. sy L˙Ž. 1 L˙Ž.0
dt t q L˙˙ y fЉ c T LL dw. Ž.HH2 Ž.1 02Lt˙ Ž. r⑀ s y r⑀ ' y r⑀ Then U111Ž .O ŽL˙ Ž .., and V ŽŽ1 x . .U ŽŽ1 x . .satisfies Ž with the dot now indicating differentiation with respect toŽ.. 1 y x r⑀ q Ј q sy Љ V¨11f Ž.R aV f Ž.cx Ž. TR 1, sy r⑀ q r⑀ lr⑀ y r⑀ V11Ž.0 Ž.S Ž.1 L Ž.2 e T Ž.1 .
At this point in the construction, AN satisfies the boundary conditions except for terms that are OŽey3 lr⑀ ., and we have ⑀ 2 Y q ANNfAŽ. N eyklr⑀ s ⑀ 2 T q U q V Љ y fЈ a q LUeylr⑀ Ý Ž.Ž.kkk⑀ ky1 1 ks2 y Ј q yl r⑀ f Ž.a RVe1 N eyklr⑀ q fЈ AUq Veyl r⑀ q T q U q V Ž.Ž011 .Ý Žkkk .⑀ ky1 ks2 1 N eyklr⑀ q fЉ Ž.AT Žq U q V . q иии .2.7Ž. 2 0 Ý kkk⑀ ky1 ž/ks1 Since eyl r⑀ Ž.fЈŽ.A y fЈ Ža q LU ., Ž.fЈ Ž.A y fЈ Ža q RV . s O , 0101ž/⑀ A SINGULAR PERTURBATION PROBLEM 685
Eq.Ž. 2.7 implies that at this stage in the approximation
y2 lr⑀ Y e ⑀ 2A q fAŽ.s O NNž/⑀ as ⑀ ª 0, uniformly on wxy1, 1 .
Having given the definitions of T11, U , and V 1, we have completed another stage in the construction of the approximationŽ. 2.4 . The remain- ing steps are similar, so we will write out only the next step in some detail. We want to eliminate all terms in Eq.Ž. 2.7 that are of size ey2 lr⑀ or ey2 lr⑀r⑀, thus
' Ј y Ј q yl r⑀ q Ј y Ј q yl r⑀ E10Ž.Ž.f Ž.A f Ža LUe . 1f Ž.A 0f Ža RVe . 1 Љ f Ž.A0 q Ž.T q U q V 2 ey2 lr⑀ . 2 111
Note that E12is even. Choose T Ž.much as in the definition of T1to satisfy
q Ј q sy ⑀ 2 lr⑀ T¨22f Ž.a ST E1e , s T˙2 Ž.0 0.
Then T2 is uniformly bounded and even, and we have
⑀ 2 Y q ANNfAŽ.
y2 lr⑀ N yklr⑀ YYee s ⑀ 22U q V q ⑀ T q U q V Љ Ž.22⑀ Ý Žkkk .⑀ ky1 ks3 N eyklr⑀ q fЈ ATq U q V y fЈ a q ST Ž.0 Ý Žkkk .⑀ ky1 Ž .2 ks2
Љ N yklr⑀ f Ž.Ae0 q 2 T q U q Veylr⑀ Tq U q V Ž.Ž.111 Ý kkk⑀ ky1 2 ks2
2 N eyklr⑀ q T q U q V q иии . Ý Ž.kkk⑀ ky1 ž/ks2
Next, we choose U22and V to eliminate from the preceding expression
ey2 lr⑀ ey2 lr⑀ fЈ A y fЈ a q ST s fЉ cx Lq RT Ž.Ž.02 Ž . ⑀⑀Ž.Ž.Ž .2 686 WALTER G. KELLEY and to make the approximation more nearly satisfy the boundary condi- q r⑀ tions. Thus we choose Ux2ŽŽ1 . . to satisfy
q Ј q sy Љ U¨22f Ž.a LU f Ž.cx Ž. TL 2, 12 U Ž.0 syT yyV ⑀ elr⑀ , 22ž/⑀⑀ 1 ž/ y r⑀ ' y r⑀ and V22ŽŽ1 x . .U ŽŽ1 x . .. Now A 2is even and satisfies the y4 l r⑀r⑀ 2 ⑀ 2 Y q s boundary conditions up to OŽe .,and A22fAŽ. OŽey3 lr⑀r⑀ 2 .. We can continue the construction in the same manner to obtain an approximation AN for any positive integer N, except at the last stage we choose UNNand V so that the boundary conditions are satisfied exactly. Furthermore, we have
yŽ Nq1.lr⑀ Y e ⑀ 2A q fAŽ.s O .2.8Ž. NNž/⑀ N
The construction of the formal approximation is complete.
3. VERIFICATION OF THE FORMAL APPROXIMATION
We will show that there is a solution ofŽ.Ž. 2.1 , 2.2 that is close to the formal approximation AN constructed in Section 2. Consider the lineariza- s r⑀ tion of Eq.Ž. 2.1 about AN Žwith x and a dot indicating differentia- tion with respect to ., ¨q Ј ¨ s ¨ f Ž.AN 0,Ž. 3.1 ¨ Ž.Ž.y1r⑀ s¨ 1r⑀ s 0.Ž. 3.2
The key step in the verification is the estimate of the size of Green’s function forŽ.Ž. 3.1 , 3.2 .
LEMMA 3.1. For sufficiently large ¨alues of N, the Green’s function GsŽ.Ž.Ž., , ⑀ for 3.1 , 3.2 satisfies
elr⑀ GsŽ., , ⑀ s O ž/⑀ uniformly on wxwxy1r⑀,1r⑀ = y1r⑀,1r⑀ for all sufficiently small ¨alues of ⑀. A SINGULAR PERTURBATION PROBLEM 687
Proof. From Eq.Ž. 2.8 , AN satisfies q s ⑀ A¨NNfAŽ.g Ž., , ⑀ s yŽ Nq1.lr⑀r⑀ N s where gŽ., O Že .. Let z A˙N ; then z satisfies q Ј s ¨˙z f Ž.AzN g, s s zŽ.0 0, ˙z Ž.0 A¨N Ž.0. ¨ Define 1 to be the odd solution of Eq.Ž. 3.1 that satisfies the initial ¨ s ¨ s conditions 11Ž.0 0, ˙˙ Ž.0 z Ž.0 . Now a standard argument based on Gronwall’s inequality can be used to show that we can make, for any ) <<< ¨ Ž3. sy Љ ¨ y Ј ¨ ) Ž.f Ž.AN Ž.1 Ž.Ž.z f Ž.AN Ž.˙1 Ž. 0 for ␦ F F r⑀. We will compute the right-hand derivative of q at zero. By Taylor’s Theorem, ¨ Ž3. 3 1 Ž.d ¨ Ž. s ¨˙ Ž.0 q , 11 3! 688 WALTER G. KELLEY where 0 - d - . Then ds ds s HH22 ␦ ¨ Ž.s ␦ ¨ Ž3. 1 1 Ž.d ¨ 220 s 1 q s 2 ˙1 Ž. ¨ 6˙1Ž.0 11 PsŽ. PsŽ. syy ¨ 22H q 2 2 2 ˙1 Ž.0 ␦ s 1 PssŽ. Ž.1 q PssŽ. 111 PP syyq ds , ¨ 22␦H q 2 2 ˙1 Ž.01␦ Ps Ž.1 q Ps s ¨ Ž3. r ¨ where PsŽ. 11Žds Ž .. Ž6˙ Ž0 .. . The right-hand derivative of q at zero is 1 qŽ. q ¨˙Ž.0 q˙q Ž.0 s lim x0 11 ␦ PsŽ. PsŽ. sq q ds . ¨ ␦ H q 222 ˙1Ž.0 0 1 PssŽ. Ž.1 q PssŽ. The expression for the left-hand derivative of q at zero, q˙yŽ.0 , is obtained by replacing ␦ with y␦. Now define ¡ q˙q Ž.0 q y ¨ for 0 - - Ž.¨ 1 Ž. ⑀ ~ ˙1Ž.0 ¨ Ž. s y 2 q˙ Ž.0 q y ¨ for y - - 0. ¢ Ž.¨ 1 Ž. ⑀ ˙1Ž.0 ¨ Then 2 is an even solution of Eq.Ž. 3.1 . We need to estimate the size of q˙qŽ.0 on the intervalwx 0, ␦ . First, y Љ ¨ q Ј ¨ P Ž.f Ž.AdN Ž.1 z f ŽAN .˙1 ssO ⑀y1 . q 2 ¨ Ž. 1 Ps 6 1Ž. Similarly, P s ⑀y2 2 O Ž.. Ž.1 q Ps2 A SINGULAR PERTURBATION PROBLEM 689 It follows that q˙˙q Ž.0,qy Ž.0 s O Ž⑀y2 ln ⑀y1 .. ¨ In order to extend 2 to the entire interval, we need to compute ¨y r⑀ ˙2 Ž.. Again, we start with Taylor’s formula, 1 ¨ Ž.Ž.Ž.Ž.Ž. s ¨˙ r⑀y r⑀ q ¨ Ž3. j y r⑀ 3 , 21 6 1 where - j - r⑀. A calculation similar to that given earlier leads to 11r⑀ Q ¨y r⑀ sy ˙2 Ž.¨ r⑀␦y r⑀ H 2 ˙1Ž. ␦ 1 q QsŽ.y r⑀ Qq˙q Ž.0 q y ¨ r⑀ ds ˙1Ž., 2 2 ¨ Ž.0 Ž.1 q QsŽ.y r⑀ ˙1 where ¨ Ž3. 1 Ž.j Qss ) 0 Ž. ¨ r⑀ 6˙1Ž. ␦ - - r⑀ ¨ r⑀ ; 2 ylr2 ⑀ ⑀ ª for s . Note that ˙1Ž.le as 0. Now Q 1 s 2 1 q Qsy r⑀ 1 2 Ž.q Ž.s y r⑀ Q and 16¨˙ Ž.r⑀ ; 1 y Ј ¨ Q f Ž.AjN Ž.˙1 Ž.j is small except for s near r⑀. In fact, 1rQ q Ž.s y r⑀ 2 is bounded away from zero, so QQ 22and 1 q QsŽ.y r⑀ Ž.1 q QsŽ.y r⑀ 2 690 WALTER G. KELLEY are bounded for ␦ F s F r⑀. It follows that elr2 ⑀ ¨y r⑀ ; D ˙2 Ž. ⑀ for some constant D, independent of ⑀. Now choose ␥ s ⑀y1 y ␦⑀Ž.so that Q Ž. ) 0 for r⑀ F F ␥, and s ¨ r¨ 2 define rŽ.1 Ž.H␥ Žds 1 Ž..s . Note that the right-hand limit of r agrees ¨ r⑀ with the left-hand limit of 2 at , and we can show as above that elr2 ⑀ rq Ž.r⑀ s O . ˙ ž/⑀ For r⑀ - - 1r⑀, define ¨ 1Ž. ¨ s r q ¨yqr⑀ y r r⑀ . 22Ž. Ž.˙˙ Ž . Ž .¨ r⑀ ˙1Ž. ¨ y r⑀ ¨ Of course, we can extend 22down to 1 in the same way, so that is ¨ an even solution of Eq.Ž. 3.1 on the entire interval. The construction of 2 implies that ¨ 2 Ž. ¨ r⑀ 2 Ž.1 is bounded for all and that elr⑀ ¨ Ž.1r⑀ s O 2 ž/⑀ as ⑀ ª 0. We now define the Green’s function in the standard way, ¨ r⑀ 1Ž.1 h ' ¨ y ¨ , 11Ž. Ž. ¨ r⑀ 2Ž. 2 Ž.1 ¨ r⑀ 1Ž.1 h ' ¨ q ¨ . 21Ž. Ž. ¨ r⑀ 2Ž. 2 Ž.1 r⑀ s y r⑀ s Note that h12Ž.1 h Ž1 .0 and that h12, h are bounded as ⑀ ª 0. Let ¨ r⑀ 1Ž.1 ␣⑀ ' . Ž. ¨ r⑀ 2 Ž.1 A SINGULAR PERTURBATION PROBLEM 691 The Wronskian for h12, h is s y s ¨ y ␣ ¨¨q ␣ ¨ y ¨ q ␣ ¨¨y ␣ ¨ W hh122121212121˙˙hh Ž.Ž.˙˙ Ž. Ž. ˙˙ s ␣ ¨¨ y ¨¨ 2 Ž.21˙˙ 12. By the properties of linear differential equations and the construction of ¨ ¨ ¨ y ¨ ¨ s ¨ ¨ y ¨ ¨ sy 22, Ž.˙˙˙˙ 1 Ž. 1 Ž. 2 Ž. 2 Ž.0 1 Ž.0 1 Ž.0 2 Ž.0 1. Then ¡ h12Ž.hs Ž. if ) s, WsŽ. ⑀ s~ GsŽ., , hsh12Ž. Ž. ¢ if s ) , WsŽ. so elr⑀ GsŽ., , ⑀ s O ž/⑀ as ⑀ ª 0, uniformly on wxwxy1r⑀,1r⑀ = y1r⑀,1r⑀ . The proof of the lemma is complete. The proof of Lemma 3.1 indicates that when we linearizeŽ.Ž. 2.1 , 2.2 about an approximation of the spike layer type, the resulting Green function is exponentially large. Consequently, even approximate solutions ofŽ.Ž. 2.1 , 2.2 with exponentially small errors, such as the ones mentioned in the Introduction that have spikes at arbitrary points in wxy1, 1 , can easily be spurious. On the other hand, if we can construct approximate solutions with sufficiently small exponential errors, then we will be able to justify the approximation, as below, with a contraction mapping argument. That the Green’s function is exponentially large is not surprising in view of earlier calculationsŽ see, for example, Wardwx 21, 22. of the eigenvalues of the differential operator in Eq.Ž. 3.1 . The second eigenvalue in the spectrum is exponentially small, so the expansion of the Green’s function in terms of the eigenfunctions should yield an exponentially large result. THEOREM 3.1. Let AN be the approximation defined by Ž.Ž.2.3 , 2.4 . Assuming the hypotheses Ž.Ž.H1 ᎐ H4 , there is, for sufficiently small ¨alues of ⑀ ) 0, a solution uŽ x, ⑀ .of Ž.Ž.2.1 , 2.2 such that for N s 0, 1, 2, . . . , eyŽ Nq1.lr⑀ uxŽ., ⑀ y Ax Ž., ⑀ s O N ž/⑀ N uniformly on wxy1, 1 as ⑀ ª 0. 692 WALTER G. KELLEY Proof. As at the beginning of this section, we define s xr⑀, with a dot indicating differentiation with respect to . Let uŽ. , ⑀ denote a solution of u¨ q fuŽ.s 0, uŽ.Ž.y1r⑀ s u 1r⑀ s 0, and define ⌽ ⑀ ' q Ž., A¨NNfA Ž.. Let ¨ ' ¨ q Ј ¨ L ¨ f Ž.AN be the linearization in Eq.Ž. 3.1 , and let ¨ ' ¨ q y ¨ y Ј ¨ N f Ž.Ž.Ž.ANNf f A be the residual nonlinear part. ' y We want to show that there is a solution u so that y u AN is small. Now such a y would satisfy Ly syNy y ⌽,3.3Ž. yŽ.Ž.y1r⑀ s y 1r⑀ s 0.Ž. 3.4 Thus we want to show that 1r⑀ TyŽ .' H Gs Ž, , ⑀ .Ž.yNy Ž. s y ⌽ Ž.sds, y1r⑀ where G is the Green’s function in Lemma 3.1, has an appropriate fixed point. Let 55и denote the sup norm for the interval wxy1r⑀,1r⑀ , and define 1r⑀ D ' 2 H GsŽ., ds 55⌽ . y1r⑀ 5555F D Let y12, y . Then 5555y F ⑀ D y Ny12Ny CŽ. y12y A SINGULAR PERTURBATION PROBLEM 693 for some bounded function CŽ.⑀ , so we have r⑀ 55y F D 1 55y Ty12Ty C H GsŽ., ds y12y y1r⑀ 2 r⑀ F 1 555⌽ y 5 2CGsH Ž., ds y12y y1r⑀ 1 F 55y y y , 2 12 if 1 55⌽ F .3.5Ž. 1r⑀ 2 4C Hy1r⑀ GsŽ., ds Also, ifŽ. 3.5 holds, then 55y F D implies 55Ty F 5Ty y T 0 5q 5T 0 5 1r⑀ 1r⑀ F CD HHGsŽ., ds55 y q GŽ. , sds5⌽ 5 y1r⑀ y1r⑀ DD FqsD . 22 By the contraction mapping theorem, we have thatŽ. 3.5 implies that T has a fixed point y with 55y F D and that y satisfiesŽ.Ž. 3.3 , 3.4 . G Since, by Lemma 3.1, there is a number N00so that, for N N , elr⑀ GsŽ., , ⑀ s O , ž/⑀ and by the calculation in Section 2, eyŽ Nq1.lr⑀ ⌽Ž. , ⑀ s O , ž/⑀ N ⑀ G Ž.3.5 is satisfied for small values of provided that N maxÄ4 2, N0 . G ⑀ Consequently, for N maxÄ4 2, N0 ,Ž.Ž. 2.1 , 2.2 have a solution uxŽ., so that eyNlr⑀ uxŽ., ⑀ y Ax Ž., ⑀ s O , N ž/⑀ Nq2 uniformly on wxy1, 1 as ⑀ ª 0. 694 WALTER G. KELLEY G q Let N be any nonnegative integer, and fix M maxÄ4 2, N 2, N0 . Then eyŽ Nq1.lr⑀ AxŽ., ⑀ y Ax Ž., ⑀ s O NM ž/⑀ N uniformly on wxy1, 1 as ⑀ ª 0. Consequently, there is a solution uxŽ., ⑀ so that eyŽ Nq1.lr⑀ uxŽ., ⑀ y Ax Ž., ⑀ s O N ž/⑀ N uniformnly on wxy1, 1 as ⑀ ª 0, and the proof is complete. 4. RELATED EXAMPLES There are a number of other problems that are closely related to the boundary value problem discussed in Sections 2 and 3, and we present some representative examples in this section. EXAMPLE 4.1. Let’s reconsider Eq.Ž. 2.1 in the case that the boundary conditions are not symmetric, ⑀ 2 uЉ q fuŽ., ⑀ s 0, uŽ.y1 s0, u Ž.1 sc ) 0. As before, we seek a solution with a spike near x s 0. Assume Hypotheses - H1, H2, H3, and H4 and that c b. Let AtN Ž.be the approximate solution constructed in Section 2 for d 2 ⑀ 2 wqŽ.1 q ␣⑀Ž.fw Ž, ⑀ .s 0, dt 2 wŽ.y1 sw Ž.1 s 0, where ␣ is to be determined. One can show that this approximation can be extended to the right so that it is defined on an interval wy1, 1 q dŽ.⑀ x, q ⑀ s ⑀ s ⑀ where ANNŽ1 d Ž..c, d Ž.O Ž., and A still satisfies the differen- tial equation to the same order as given in Section 2. As in Section 3, there exists a solution wxŽ., ⑀ so that eyŽ Nq1.lr⑀ wtŽ., ⑀ y At Ž., ⑀ s O N ž/⑀ N A SINGULAR PERTURBATION PROBLEM 695 uniformly on wy1, 1 q dŽ.⑀ x as ⑀ ª 0. Make the change of variable 2t y dŽ.⑀ x s , 2 q dŽ.⑀ and define uxŽ., ⑀ s wtw ŽŽ.., ⑀ . Then u satisfies 1 ⑀ 2 uЉ q Ž.Ž.1 q dŽ.⑀ 1 q ␣⑀ Ž.fu Ž, ⑀ .s 0. 2 If we choose 1 q ␣⑀Ž.s 2r Ž2 q d Ž..⑀ , then ␣⑀ Ž.s O Ž.⑀ and ux Ž., AxN Ž.are the desired solution and approximation. Note that the change of the right boundary value from 0 to positive shifts the position of the spike to the left by OŽ.⑀ . EXAMPLE 4.2. Consider again the boundary value problemŽ.Ž. 2.1 , 2.2 , ⑀ and now seek a solution with two spikes. Let AtN Ž., be the approxima- tion of d 2 w ⑀ 2 qŽ.1 q ⑀Ž.fw Ž, ⑀ .s 0, dt 2 wŽ.y1 sw Ž.1 s 0, - - X ⑀ s constructed in Section 2. Choose 0 x0 1 so that AxN Ž.0 , 0. De- wxy q fine BNNto be the extension of A to the interval 1, 2 x0 1 obtained s by reflection of AN in the line x x0. Then BN has two spikes that are wxy q evenly spaced and is an approximate solution on 1, 2 x0 1 . Define wxŽ., ⑀ to be the corresponding solution. We can simply rescale and make a judicious choice of ⑀Ž.Žas in Example 4.1 . to obtain the desired approximation and solution with two spikes forŽ.Ž. 2.1 , 2.2 . Solutions with three or more spikes can be obtained by similar patching and rescaling. EXAMPLE 4.3. Finally, we consider a boundary value problem intro- duced by Carrierwx 2 that has often been associated withŽ.Ž. 2.1 , 2.2 , ⑀ 2 uЉq21Ž.y x 2 u q u2y1 s 0, uŽ.y1 su Ž.1 s 0. The basic issues are whether this problem has solutions with spikes and where in the interval do these spikes occur? Bender and Orszagwx 1 showed that formal perturbation analysis leads to the conclusion that it is consis- tent to have any number of spike layers at any location in wxy1, 1 . The stable reduced solution for this problem is 2 axŽ.' x 22y 1 y '1 q Ž.1 y x . 696 WALTER G. KELLEY y r⑀ Define SxŽŽx0 . . to be the solution of ⑀ 2 Љ q y 2 q 2 y s S 21Ž.x0 S S 1 0, x y x ª 0 ª "ϱ S 0as ⑀ . Ž.S is constructed as in Section 2. Now define u JuŽ, x .' H 21 Žy x 2 .q 2Ž. q ax Ž. aЈ Ž.x y 4 x Ž. q ax Ž. d . 0 A short calculation yields 2 xŽ.1 y x 2 JuŽ., x s ua22Ž.Ј Ž.x y 2 x s u . 2 '1 q Ž.1 y x 2 Kathwx 7 employed the Melnikov integral to conclude that a necessary s condition for the appearance of a solution with a spike at x x0 is ϱ 2 x 1 y x 2 ϱ s y r⑀ s 00Ž. 2 y r⑀ 0 HHJSxŽ.Ž.x00dx SxŽ.Ž.x dx, yϱ q y 2 2 yϱ '1 Ž.1 x0 s " wx which vanishes at x0 0, 1. 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