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 <p,ip EU, we obtain \\F(<p) - F(n\H~(n) a = || - E »r\\ = || Hm(n) n=0 n=0 * ' n=l oo < J2 K\K\\<P - n H-M y*-1 +... +vn_1u H^n) n=l oo < \o-n\Kn~ln sup Nl^) \\<p - i>\\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))<p(x) for v,tp € Hm(i2). Proof. Let v G Hm(fl) be a fixed element. We will show (4.2) ||F(t, + <p) - F{v) - F'(v)<p\\Hm{n) = o(M|i/.n(i2)). 598 J. Napiorkowska Observe that the formal derivative of F has the form oo 1 F'(v) = ^na^"- e Hm{!7), n=l if V € Hm{fl).