DEMONSTRATE MATHEMATICA Vol. XXXIV No 3 2001
Joanna Napiôrkowska
REGULARITY OF SOLUTIONS OF NONLINEAR PARABOLIC EQUATIONS
1. Introduction One of the main topics considered by mathematicians dealing with evo- lution partial differential equations is the problem of regularity of solu- tions. This problem was investigated by many authors, in particular by A. Friedman [FR], O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural- ceva [L-S-U], S. Smale [SM], C. Bardos [BA]. In the context of semi-linear evolution equations of parabolic type, the regularity was studied, e.g., by R. Temam [TE]. Consider a nonlinear evolution problem of the form ù + Au = F(u), t > 0, (1.1) u(0)I = u0. In this paper we study the smoothness of solutions for the Cauchy prob- lem of the type (1.1) with A being an abstract counterpart of an elliptic op- erator. Our main result is a necessary and sufficient condition of the higher order regularity of solutions for such nonlinear partial differential equation. In Section 2 we introduce some preliminaries. Section 3 is devoted to the regularity result of Temam [TE] in the case of linear equations. Section 4 contains the main result of the present paper. Finally, the applications and examples are described in Section 5.
2. Preliminaries Let if be a real Hilbert space with scalar product (•, •) and the norm || • ||. Let (A,D(A)) be a linear operator acting from D(A) into H, where D(A) C H is the domain of A. Assume that (A,D(A)) is a positive (i.e., (Au, u) > 0 for all u € D(A) \ {0}) and self-adjoint (see, e.g., [PA]) operator with D(A) dense in H. Moreover, assume that {A, D(A)) is an isomorphism 594 J. Napiorkowska from D(A) onto H such that its inverse {A~x,D(A~l) = R(A)) is a linear, compact operator in H. Denote by cr(A) the spectrum of A. Because A is a self-adjoint and positive operator, its spectrum is contained in a positive real axis. Since Re cr(A) > 0, we can define fractional powers of operator A (cf. [CH-D], [FR], [HE], [PA]).
Now, use symbol Va for the domains of , a G M. Two spaces are particularly important in our further considerations; V = V\ called the energetic space and the basic Hilbert space H — VQ. For a > 0 the space V_a can be identified with the dual (Va)' of space Va. Besides the scale of Hilbert spaces VA, A G R, consider a family of Hilbert spaces EM, m G N, having the following properties:
(2.1) EM+1 C EM for any meN, the injection being continuous, EQ = H, (2.2) Vm is a closed subspace of Em, meN, the norm induced from Em on Vm being equivalent to the norm of Vm.
Usually, Vm is a subset of Em consisting of all functions, which satisfy appro- priate boundary conditions. For example, for m = 2 and the Sobolev space 2 2 E2 = H {Q) we will take V2 = {u G E2 : u - 0 on dQ} = H%(i2) n H {Q). Consider further a linear differential operator A of the order 2r, associ- ated (see (2.5)) with the operator A, and such that:
(2.3) A is a continuous map from EM+2 into EM for M > 0, (2.4) A is an isomorphism from Em+2 H D(A) onto Em for m > 0, (2.5) Au = Au for all u G Vm, m > 2. By A we mean a differential operator, defined on sufficiently smooth func- tions, while (A,D(A)) is an abstract operator, associated to A that ex- tends A. Validity of the condition (2.4) for elliptic operators is discussed in [TR, p.490, Theorem 5.5.1(b)].
The spaces Em and Vm are of particular importance for the studies of elliptic equations. Similarly, in the case of evolution equations, we define the spaces WM]
Wm = ju G C([ 0, T]; £m) : ^ G C([0, T]-Em.2j)J = 1,..., Z j for I = [y], m G N. The derivative ^ is a strong derivative (that means the limit of difference quotient in the space H). The following result holds: LEMMA 2.1. For u G ^«+2+2*;, s G N, k > 1, we have d / d*-xti\ _ dku Jt\Al^) ~Adt Nonlinear parabolic equations 595 where the time derivatives are understood in the sense of vector-valued dis- tributions (cf. [LI, p. 7]). The above identity was used, e.g., by Temam [TE] to compute derivatives of the solution u of linear problem.
3. Linear theory Following R. Temam [TE], consider the initial value problem
(3.1) \di + = <>0'
{ u(0) = it0. The following regularity result is well known:
FACT 3.1. Let T > 0, 2 (3.2) f € L (0,T-,V') and u0<=H. Then the problem (3.1) possesses on [0,T] an unique solution u in (3.3) L2(0,T; V) fl C([0,T]; if). If furthermore 2 (3.4) f€L (0,T-,H), u0eV, then (3.5) ueL2(0,T-,D(A))nC([0,T}-,V). The proof of this result can be found for instance in the monography by J. L. Lions and E. Magenes [L-M, Theorem 4.1]. For the linear problem (3.1) R. Temam stated the following regularity result (see [TE]) for a second order operator A (r = 1). THEOREM 3.1 (R. Temam). Assume that the hypotheses of Section 2 are satisfied. Moreover, suppose that for some m> 2
u0eEmnD(A), f € Wm-2, and dfj (L2(0,T-,H) if m = 21 + 1, dtl \L2(0,T-,V') ifm = 21. Then a necessary and sufficient condition for a solution of (3.1) (satisfying condition (3.3)) to belong to Wm is that
~^(0)ev for j = 1,...,/ — 1,
dlu ifm = 2l + l, dtl K ' \H if m = 21, 596 J. Napiorkowska where
t=0 v 7 for j = 1,...,1 (l = [?]). The proof of this theorem can be found in [TE, p. 79].
4. The main result As a consequence of the linear theory, we are able to obtain the re- sult concerning nonlinear problems and extending considerations to the case of higher order elliptic operators. We illustrate our Theorem 4.1 with two exemples. Assume that i? C Mn is a bounded domain of class C°° and denote by EQ a Sobolev space; EQ = HM°(F2) with mo > f. It is well known that in this case Hm°(i2) is a Banach algebra (cf. [AD, p.115], [TE1, p.51]). The latter means that (4.0) there exists a constant K depending only on Q such that if u,v € mo mo H (i2), then uveH (i2) and |M|Hm0(fl) ^XIMlHmo^jIMItfmo^). The following remarks will be useful in the sequel:
Lemma 4.1. Let f : R —• M be a function defined by a power series centered at 0 with infinite radius of convergence. Then the Nemytskii operator
F(v)(x) = f(v(x)) for v € Hm{i2), m > m0 > §, maps Sobolev space Hm(i1) into itself. Proof. Let / be a function defined by a power series centered at 0 with infinite radius of convergence
(4.1) /(y) = f>nyn, -ir=Tta"VM = o. ' * XL n—>oo 71=0 Let v G Hm(Q) be a fixed element. Consider the function series oo n F(v) = Y,anv . n=0 By (4.0) we have the following estimates:
OO 00 £ KIIKIlH-ifl) = |oo| + £ n=0 n=l oo ^laol + ^Kl^-1^!!^). n=l Nonlinear parabolic equations 597
Taking
IMItfm^) < N0 for fixed v € Hm{f2), we get oo oo J2 KI^IMI^) < £ Kiff-jvy < oo. n=l n=l oo By (4.1) the power series an'rn is convergent, with infinite radius of n=l convergence. Therefore, it is convergent for r = KNq. Hence the considered function series is absolutely convergent in Hm(f2)-norm, which shows that F(v) e Hm{fi) whenever v G Hm(f2).
LEMMA 4.2. Under the assumptions of Lemma 4.1, the function F : Hm(f2) —>• Hm(ii) is continuous and F maps bounded subsets of Hm(f2) into bounded subsets of Hm(fl). More precisely, F is Lipschitz continuous on bounded subsets of Hm(f2).
Proof. Let U C Hm(f2) be a bounded set. For \\F( a = || - E »r\\ = || Hm(n) n=0 n=0 * ' n=l oo < J2 K\K\\ < \o-n\Kn~ln sup Nl^) \\ \\Hm(n). The convergence of the last series follows from (4.1) with the coefficient |an|n. Indeed, lim v/|on|n = lim irflaJ = 0, n—»oo n—»oo so that the considered series has the same radius of convergence as (4.1). Therefore F is Lipschitz countinuous on bounded subsets of Hm{Q). REMARK 4.1. Since m > F, then Hm((2) C L°°{Q). Therefore bounded sets of Hm(f2) are bounded also in L°°(f2). LEMMA 4.3. Under the assumptions of Lemma 4.1, the function F : Hm(f2) Hm(Q) has Frechet derivative F' € C(Hm(i2),Hm(i2)) given by [F»¥>](x) = f'(v{x)) Proof. Let v G Hm(fl) be a fixed element. We will show (4.2) ||F(t, + Observe that the formal derivative of F has the form oo 1 F'(v) = ^na^"- e Hm{!7), n=l if V € Hm{fl). Moreover for F'(v) \\F(v + = £ On((t> + oo n / v m n=2 fc=2 v ' H {ri) i-k "—n=22 /c—fc=22 Notice that for fc = 2,..., n and all n > 2. Therefore, we obtain n=2 fc=2 ^ ' < E ^-Vni ¿n2 (" _ f) inir-WMiW) n=2 fc=2 V ' J^-2 £ tfn-1,^2,,^2^ g (n - A \\vrH~^)M^m{n) n—2 j=0 n=2 n 2 n 2 < IMIlr-(fl) E^ |an|n (||t;||H^) + IM|i/-(rt)) - n=2 S= 2 8+2 2 =" IMUr-«» E^ |as+2|(5 + 2) (||U||ifm(n) + Mh-wY. *=o Nonlinear parabolic equations 599 s+2 s The sum 0K \as+2\(\\v\\Hm^n) + |M|tfm(0)) is finite. Whence the condition (4.2) holds. Therefore F is differentiable in the sense of Fréchet. More generally, from Lemma 4.3, we get: LEMMA 4.4. Under the assumptions of Lemma 4.1, the function F : Hm{Q) —> Hm{Q) has Fréchet derivative of any order k, k € N and F^ : Hm(Q) -» Hm{(2). Consider now the problem (1.1) we introduce the following assumption. ASSUMPTION 4.1. Let í? C M" be a bounded domain of class C°°. Let H = m m +mr H °(Q) for some ro0 > f and Em = H ° (i2) for m > 0. Let (A, D(A)) be an elliptic operator of the order 2r, giving a sectorial operator (cf., e.g., [HE, Definition 1.3.1, Examples, p. 19], [CH-D, Chapter I, Section 3], [FR, Part 1, Section 19]) with a domain D(A) C H. As a Nemytskii operator (cf., e.g., [C-H, p. 34]) we consider the function F : Hm{Q) Hm(i2), m > mo > associated with / : R —> R. The function f has a form of power series with center 0 and infinite radius of convergence. Thus, as in Lemma 4.1, F will be written in the form F(u)(:r) = f(v(x)) for v £ Hm(Q). REMARK 4.2. The domain of A depends on the choice of the base space H. The domain of A in L2{Q) is the set Dq(A) = {« G H2t(Q) : boundary conditions are satisfied}. If we consider the operator A in the space Hm(f¿), m > mo > j, then the domain of A is m+2r H (f2)nD0(A). This characterization of the domain of A shows that A is an isomorphism from the domain of A onto H and the domain of -A is a subset of H. Denote by Xa the domain of Aa, a > 0; in particular H = Xo and D(A) = X1. Then we can define a local Xa-solution of (1.1) (see, e.g., [CH-D, Chapter II], [HE, Section 3.3]). a a DEFINITION 4.1. Let u0 € X , a e [0,1). A function u e C([0,T]; X ), for some real T > 0, satisfying the following properties: (i) u(0) = it0, 1 (ii)ti€C ((0>r];#), (iii) u(t) belongs to D(A) for each t REMARK 4.3. As was shown in Lemma 4.2, the function F is Lipschitz continuous on bounded subsets of Hm(f2). Since Hm(ii) c Hm°(Q), m > mo > f , then bounded sets of Hm(f2) are bounded also in Hm°(i2). We will show below that F is Lipschitz continuous as a map from Xa, a G [0,1), into H = Hm(fi), m > mo > For simplicity, we will not distinguish ( in the rest of the paragraphs ) the dependence of Xa on the base space Hm(fi). Formally we shall denote 1°. Let U be a bounded subset of Xa, a 6 [0,1). For tp,ip €U, we have IIFfo) - F{4OH* < Cu\\ u G C((0,T];X^),7 G [0,1). In particular, for a = ^, we get the solution such that (4.3) ueC(l0,T];V)nC1((0,T];H) and , e.g., u G C((0,T]; V), where V - X?. Additionally, we will show that (4.4) ue L2{0,T-,D(A)) for an Xa-solution of problem (1.1). This additional smoothness of the local Xa-solution will be needed in the sequel. Multiplying equation (1.1) by Au in H and then integrating over the interval (0, t), we get \ f^,Au \ dr + jj (Au, Au) dr = j| (F(u), Au) dr, T / o o where (•, •) is the scalar product of Hilbert space H. Moreover, from the Cauchy-Schwartz inequality, we obtain 2 |(F(uMu)dr < j IIF^IUMuII^t < \ j ||F(u)|& dr + ^ j \\Au\\ Hdr. 0 0 0 0 Therefore 2 J (%Au) dr + \\ \\Au(r)\\ldr < \ j \\F(u(r))\\ H dr < Ct, 0 ^ ' 0 0 Nonlinear parabolic equations 601 where C is a constant, because u G C([0, T]\H) and F maps bounded subsets of H into bounded subsets of H. Moreover, for a positive and self-adjoint operator A in a Hilbert space H, we have | {tA") dT = j (AiTr-" j Ai") * = S\i-Ulu{T)f„iT = i||4il.(0||l, - i||AlUo||i,. 0 Collecting all the estimates, we find that 2 2 2 ±||A*u(i)|| * + \ i \\Au(T)\\ HdT < l\\Aiuo\\ H + Ct for t € [0,T]. Therefore the estimate (4.4) holds and moreover ueL°°(0,T;V), provided that (4.5) u0 e £>(>1). Consider a linear equation of the form dfiL (4.6) —+A(t)u = f(t) on (0 ,T) with the initial condition (4.7) it(0) = «o. For (4.6) we consider a family a(t; •, •) of bilinear continuous forms on V, t G [0,T], associated with the linear operator A(t), setting a(t;u,v) = (A(t)u,v)v,v for u,vGV. Let these forms have the following properties: (i) For every u, v G V, t a(t; u, v) is a measurable function, (ii) For a.e. t G [0,T], a(t;u,v) is a bilinear continuous form on V, i.e., there exists 0 < Mt < oo such that |a(i;u,u)| < Afr||u||v||v||v, for every u,v G V, a.e. t G [0,T], (iii) For a.e. t G [0,T], a(t\u,v) is coercive, i.e., there exists a > 0 such that a(t;u,v) > a||tt||^, for every u G V, a.e. t G [0,T]. Then, from the Lax-Milgram lemma, for some t G [0, T] the operator A(t) defines an isomorphism from V onto V. It gives also an isomorphism from D{A(t)) onto H. 602 J. Napiorkowska REMARK 4.5. If the form a is not coercive, then we multiply (4.6) by e~mt (m is some constant) and consider the new form associated to A(t) = A(t)+ml. Note that this operation does not change the smoothness of solution. Let us recall the following fact: PROPOSITION 4.1 [TE2, Theorem II.3.4]. Let the assumptions (i)-(iii) hold. 2 For given u0 € H and f G L (0,T;V') (or L?(Q,T\H)), there exists a unique solution u of (4.6)-(4.7) such that 2 2 u G C([0,T];ii)nL (0,T;y)1 ^ G L (0, T; V'). We have the following result similar to Theorem 3.1. Instead the func- tion / depending on t (cf. Section 3), we consider now a function F depend- ing on the solution u. THEOREM 4.1. Let Assumption 4.1 be satisfied. Suppose that (4.8) u0 e Eii n D{A). Then a necessary and sufficient condition for X? -solution of the nonlinear problem (1.1) to belong to W2i is that dUi (4.9) ^J(0)Gfi f°ri = 1>--->1- Proof. Let us fix 21 > mQ > j. We first prove the necessity of condition (4.9). Assume that u G W2i. Then we have u G C([0,T]-,E2L), f G C([0,T];E2L_2), .., ^ G C([0,T]; Eq). In particular we have J d u „ r ^-(0) G E2l-2j for j = 1,... ,1, when (4.8) is satisfied. Since E2i-2j C H, we obtain the result. We shall now prove sufficiency of condition (4.9). Suppose that the con- dition (4.9) is satisfied. Then ^(0) G H for j = 1,..., I. We will show step by step, from j = 1 to j = I — 1, that -solution of (1.1) belongs to W21, i.e. G C([0,r]; E2l_2j) for j = 0,..., I. Because of (2.5), the equation (1.1) can be rewritten as (4.10) ^ + Au = F(u). at We start with j = 1. Differentiating (4.10) with respect to i, we obtain d , . . d r„. Nonlinear parabolic equations 603 where the time derivatives are understood in the sense of distributions (cf., e.g., [FR, p.11]). By Lemma 4.3 derivative F(u)] exists and it is equal to F'(u(t))4j±. Moreover, applying Lemma 2.1, we can write /„ , .x d2u ,du , ..du This equation is linear for jjf. Replace ^f by U\. Then we get (4.12) and we can consider (4.12) as an linear equation of the form (4.13) ^ + .4(^ = 0 with the linear main part operator A(t) = A-F'(u(t)). The initial condition now reads as U1(0) = ^(0) = -Au(0) + F(u(0)). This way we get the particular problem of the form (4.6) that was studied by R. Temam (cf. [TE1, p.72]). From (4.9) we have ^(0) = t/i(0) G H. The bilinear form a(£;-,-), associated with A(t), satisfies the conditions (i)-(iii) required in Proposition 4.1, and this allows to apply Proposition 4.1 to the equation (4.13). Whence Ui G C([0,T]-,H), which shows that % € C([0,r];ii). Using (4.10), we get du Au = -—+F(u)eC([0,T]-,H), since F(u) G C([0,T]-,H). Hence, by (2.4), we find u G C([0,T]; E2). Note that, if n = 1 and 21 = 2, then ^(0) G H and from Proposi- tion 4.1 it follows that ^ G C([0,T]\H). Futhermore, by (4.10) and for F(u) G C([0,T]; H) we have Au G C([0,T]-,H). Again from (2.4) we obtain u G C([0,T]; E2), whence u G W2. In the second step of recurrence, differentiating (4.11) with respect to t, we obtain a linear equation for ^ Writing U2 = ^¡t , we get (4.14) 604 J. Napiorkowska Hence, we again obtain the equation of the form (4.6) which was studied by R.Temam. Since F"(u(t)) ^jy(O) = C/2(0) € H. The main part of the equation for U2 is the same as in the equation (4.13). Applying Proposition 4.1 to (4.14) we deduce that U2 ,du (Pu . s.du ^,r since F'(u(t)) e C([0,T];#). Then, by (2.4), % G C([0,T];£2)- We shall show that u <= C([0,T];E4). Indeed, du . Au = -— + F{u) at and P € C([0,T];£2), F(U) € C([0,T];£2) for u 6 C([0,T];£2) . Similar reasoning will be continued for j = 3,..., I — 1. Notice that the smoothness of derivatives of the solution u increases in every step. Generally, after j — 1 steps, we have j+1 j j d u ,d u d r , X1 (4'15) difM + = ' for j = 1,..., I, which was obtained by differentiating j times the equation (4.10). By Lemma 4.4 the derivative $j[F(u)] exists and has the form ,4,6, >.)]-£i*>«0)Ea.( 4=1 a s ' x ' where a = (a 1,..., ctj), ai + ... + otj — s, lax + • • • + jotj = j• The coef- ficients Csa in (4.16) are given by the recurrent formula; for the derivative F^(u) we set C 1 ( ia = and C £ = kCii^ + CliZl]a. Moreover, if j < k, then cjf^ — 0. Using (4.16) the equation (4.15) can be rewritten as = £F('>W<))£a..• • (¿tFi s=2 a' v ' x where a' — (ai,... ,aj_i), ai + .. . + ctj-i = s, lai + ... + (j - 1)0,-1 = j• Nonlinear parabolic equations 605 The coefficients Csa' are given similarly to Csa. We set x x x N s=2 a' ' Denote by Uj the derivative then (4.17) + A(t)Uj = R(F^(u(t)), for j = 1,..., I. The above equation again has the form (4.6). The derivative F^(u(t)), s = 2,...,j, is continuous in H by Lemma 4.4 and from the smoothness property of -X"5-solution (see (4.3)). In addition, the sum £(^)Ql... (f^r)^-1 is a' continuous in H as a consequence of the previous steps and the multiplicative property of a Banach algebra. Therefore R(F^u(t)), ^j^r) G C([0,T];H). Applying Proposition 4.1 to (4.17) we obtain Uj € C([0, T\;H), which shows that^eC([0,T];ii)forj = l,...,Z. In the last step, talcing (4.17) with j = I — 1 and using the earlier con- siderations, we find that ^ G C([0,T];H), G C([0,T]; £2),... ,u G C([0, T}; E2i). It shows that u G W2i- The proof of Theorem 4.1 is complete. REMARK 4.6. An interesting question would be an extension of Temam's result to the case of nonlinear equations with higher order elliptic operators. REMARK 4.7. The case of m odd can also be considered. In that case, if X 2-solution of (1.1) belongs to W2/+1, then (13 iti ^-(O)GF for j = l,...,l. We omit the proof of the last fact, since the argumentation is analogous. 5. Applications and examples In this section we describe some nonlinear evolution equations, which arise in chemistry, physics and mechanics, that will serve as examples illus- trating Theorem 4.1. Most of the nonlinear problems are derived from their physico-chemical context. Consider exothermic chemical reactions in combustible materials. The interaction of the chemistry of the species with the basic fluid flow is described by a highly nonlinear, extremely complex, degenerate, quasi- linear parabolic system of partial differential equations. Combustible sys- tems composed of gases, liquids, or solids can experience thermal reaction 606 J. Napi6rkowska processes, which axe typically initiated by boundary heat addition, by local- ized volumetric heating, by the passage of a dynamic wave, or by very fast compression. Consider the following problem 'ii = Au + eu, t> 0,xeii, (5.1) u = 0 on dfi, tt(0) = ito in Q. This problem arises in the theory of thermal self-ignition of a chemically ac- tive mixture of gases in a vessel. This reaction was mentioned by I.M.Gelfand in 1959 (see [GE]). Therefore this model is referred to as the Gelfand prob- lem. Another examples occur in mathematical biology, physics and chemistry as reaction-diffusion systems (see [TE1]). Return to Theorem 4.1. We first study an equation associated to Laplace operator (r = 1); assume that Q C Mn is a bounded domain of class C2. EXAMPLE 5.1. Consider the Dirichlet problem ' u — Au = f(u), x £ f2, t> 0, (5.2) u = 0 on dfi, u(0,x) = uo(x), x G fi. For example, the nonlinear function / may be given as: (i) f(u) = eu (the solid fuel ignition model) , (ii) /(u) = sinu, (iii) f(u) = W(u), where W(u) is a polynomial depending on the solution u. 3 A For n = 4 we set m0 = 3 > f, H = H (i2), V = {u G H {f2) : u = 0 in the sense of traces on 0 and 5 Au = -Au for u G H {Q) n V. All asumptions of the paper concerning the spaces and the operators are satisfied by these special examples. It particular, —A is an elliptic operator giving a sectorial operator (cf. [CH-D], [HE], [PA]). For I = 1, if uo G D(A), then the Dirichlet problem possesses X'-solution u which belongs in particular to C([0,T]\ D(A)). For I = 2, assuming (4.8), the necessary and sufficient condition for u to be in C([0,T]\H7(f2) fl V), is that Au o + f(uo) = 0 on dQ. We next study a biharmonic equation {r — 2). We assume that i?cRn (n < 3) is a bounded domain of class C4. Nonlinear parabolic equations 607 EXAMPLE 5.2. Consider the Cahn-Hilliard problem u = A(-Au +f(u)), x G i2, t > 0, { du d(Au) n u(0,x) = UQ(X), x€i2, where N is the unit outward normal on dfi and / is a polynomial of odd order with positive leading coefficient. 2 4 For n = 3 we set m0 = 2 > §, H = H {Q) and V = {u G H {Q) : = 0 in the sense of trace on df2}, (A, £>(>!)) = {u G H6(/2) : = 2m+2 ^jfii = 0 in the sense of trace on di2}), Em = H (i2) for m > 0 and Au = A2u for u G H6(f2) n V. All asumptions concerning the spaces and the operators are satisfied. In particular, A2 is a sectorial operator (cf., e.g., [CH-D, p.40]). As a consequence of Theorem 4.1, for I = 1, if uo G D(A), then the so- lution u of the Cahn-Hilliard problem belongs to C([0, T];D(A)). For I = 2, assuming (4.8), expression 2 -A u0 + Af(u0) = 0 on dQ is the necessary and sufficient condition for u to be in C([0, T]; Hw(i2)n V). 6. Comments Imposing additional restrictions on the initial data uo , such as smooth- ness requirements and compatibility conditions, we obtain, through Theo- rem 4.1, extra space regularity of the solution u(t) for all admissible values t > 0. Proposed approach allows to get arbitrarily high regularity of the local in time solutions to (1.1) provided necessary compatibility conditions hold and sufficiently regular initial data UQ is considered. There are a lot of possible applications of the above results. The main examples here are the solid fuel ignition model called Gelfand problem and the Cahn-Hilliard pattern formation model. References [AD] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 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