Asymptotic behaviour of nonlocal reaction-diffusion equations
Abstract
The existence of a global attractor in L^2(Ω) is established for a reaction-diffusion equation on a bounded domain Ω in R^d with Dirichlet boundary conditions, where the reaction term contains an operator F : L^2(Ω) → L^2(Ω) which is nonlocal and possibly nonlinear. Existence of weak solutions is established, but uniqueness is not required. Compactness of the multivalued flow is obtained via estimates obtained from limits of Galerkin approximations. In contrast with the usual situation, these limits apply for all and not just for almost all time instants.
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Asymptotic behaviour of nonlocal reaction-diffusion equations∗ Mar´ıa Anguiano Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160 41080 Sevilla, Spain E-mail: [email protected] P.E. Kloeden and T. Lorenz Fachbereich Mathematik, Johann Wolfgang Goethe Universit¨at D-60054 Frankfurt am Main, Germany E-mail: kloeden / [email protected] Abstract The existence of a global attractor in L2(Ω) is established for a reactiondiffusion equation on a bounded domain Ω in Rdwith Dirichlet boundary conditions, where the reaction term contains an operator F:L2(Ω) → L2(Ω) which is nonlocal and possibly nonlinear. Existence of weak solutions is established, but uniqueness is not required. Compactness of the multivalued flow is obtained via estimates obtained from limits of Galerkin approximations. In contrast with the usual situation, these limits apply for all and not just for almost all time instants. 1 Introduction A simple population model with spatial dependence is given by the reactiondiffusion equation ∂u ∂t = ∆u+u(1 −u) , (1) on a bounded domain Ω in Rdwith Dirichlet boundary conditions. The long term dynamics of this model is well understood. In the above model the population at a point x∈Ω depends only on its value at this point, apart from the diffusivity term. More realistically, it could ∗Partially supported by the Ministerio de Ciencia y Tecnolog´ıa (Spain) and FEDER (European Community) grant BFM2002-03068 as well as the DFG grants KL1203/7, LO273/5) 1
depend on the population size at other points, in particularly at near by points. For example, it could depend on the average over a small neighbourhood uδ(t, x) = ZB(x;δ) u(t, y)dy ZB(x;δ) dy , for some small δ > 0 instead of on u(t, x) itself. This leads to a nonlocal PDE ∂u ∂t = ∆u+uδ(1 −uδ) . (2) Alternatively, uδ(t, x) could be some other functional of the solution u(·, t) evaluated at the point x. There are many applications of nonlocal effects in partial differential equations in the literature, e.g., in combustion theory [13] and the Navier-Stokes equations [5]. The nonlocal term is often an integral operator and the equations are then called “integro-differential” equations. Boltzmann equations are a very well known class of integro-differential, but are first order unlike those of interest here. However, the nonlocal term could be different, see, e.g., the review article by Bates [3]. There is a large literature on the existence, regularity and blow–up of solutions of nonlocal evolution equations, see for example [15, 16, 17] and the papers cited therein. Global attractors for nonlocal evolution equations have been investigated recently for the globally modified Navier-Stokes equations by Caraballo et al. [5], for m-Laplacian parabolic equations with a nonlocal nonlinearity by Chen [6] and by Hilhorst et al [8] for a nonlocal Kuramoto–Sivashinsky equation. Several aspects of reaction-diffusion equations are being analyzed over the last years, particularly, their asymptotic behaviour, see for example [14,15] and [17]. In this paper we consider general nonlinear nonlocal terms in autonomous reactiondiffusion equations, which generate strict multivalued semiflows. In particular, we establish the existence of a global attractor after first proving weak solutions and the compactness of attainability sets of the multivalued semiflow. For this we use estimates obtained as limits of Galerkin approximations which hold for every time instant and not just for almost all time instants. The problem is formulated in the next section and dissipativity estimates are presented in Section 3 with some longer proofs given at the end of the paper in Section 7. The existence of weak solutions is established in Section 4, while the generation of a strict multivalued semiflow and the existence of a global attractor are shown in Section 5. Finally an explicit example is presented in Section 6. 2 Setting of the problem Let Ω ⊂RNbe a bounded open set, it satisfies the Poincar´e inequality, i.e., there exists a constant λΩ>0 such that ZΩ u2(x)dx ≤λ−1 ΩZΩ (∇u(x))2dx,∀u∈H1 0(Ω).(3) 2
Let (·,·) denote the scalar product in L2(Ω) and k·kL2(Ω) the corresponding norm in L2(Ω). In addition, let h·,·i denote the duality product between spaces H1 0(Ω) and H−1(Ω). Consider the following initial boundary value problem for a nonlocal reaction– diffusion equation with zero Dirichlet boundary condition in Ω, ∂u ∂t +A u =F(u) in Ω ×(0, T), u= 0 on ∂Ω×[0, T], u(x, 0) = u0(x),for x∈Ω, (4) where Ais a uniformly parabolic operator in divergence form with aij =aji ∈ L∞(Ω), 1 ≤i, j ≤N, for which there exist constants λA, ΛA>0 such that λA|η|2≤ N X i,j=1 aij(x)ηiηj≤ΛA|η|2 and Au(x) := − N X i,j=1 ∂xjaij(x)∂xiu(x)(5) for all x∈Ω and η∈RN. Remark 1 The operator induced by Acan be interpreted as A∈ LH1 0(Ω), H−1(Ω) and is symmetric with hAv, vi ≥ λAk∇vk2 L2(Ω) for all v∈H1 0(Ω). Since H1 0(Ω) is included in L2(Ω) with compact injection, as a consequence of the Hilbert-Schmidt Theorem there exists a nondecreasing sequence of positive real numbers, 0< λ1≤λ2≤... ≤λk≤......, with limn→∞ λn= +∞and there exists an orthonormal basis {wk:k≥1}of L2(Ω). Moreover, {wk:k≥1}is an orthogonal basis of H1 0(Ω) with A wk= λkwkfor all k≥1, where (u, v)H1 0(Ω) Def. =hAu, vi. The operator F:L2(Ω) →L2(Ω) fulfills the following assumptions: a) Fis continuous with respect to the L2norm. 3
b) there exist β∈(0, λAλΩ) and Cβ>0 with (u, F(u)) ≤βkuk2 L2(Ω) +Cβfor every u∈L2(Ω),(6) c) there is some nondecreasing Ψ : [0,∞)→Rsuch that for all u∈L2(Ω), kF(u)k2 L2(Ω) ≤ΨkukL2(Ω)(7) In special cases Fwill also be assumed to satisfy a local Lipschitz condition: d) For all R > 0 there exists LRsuch that if v, w ∈L2(Ω) with kvkL2(Ω) ≤R, kwkL2(Ω) ≤R, then kF(v)−F(w)kL2(Ω) ≤LRkv−wkL2(Ω) . (8) Remark 2 In the subsequent statements and proofs, condition (c) can be easily replaced by the slightly weaker assumption that there is a nondecreasing function e Ψ : [0,∞)→Rsuch that for every u∈H1 0(Ω) with Au ∈H1 0(Ω), |hF(u), A ui| ≤ e ΨkukL2(Ω).(9) kF(u)k2 L2(Ω) ≤e ΨkukH1,2(Ω).(10) 3 Dissipativity estimates Both the existence of weak solutions and of bounded absorbing sets in L2(Ω), H1 0(Ω), respectively, are based on the following a priori estimates. Their proofs do not require the uniquenss of weak solutions for a given initial value. Some of the auxiliary results are formulated for Galerkin approximations, and their (quite technical) proofs are postponed to §7. Proposition 3 If Fsatisfies hypothesis (b), then every weak solution u∈ L20, T;H1 0(Ω)of (4) with u0∈L20, T;H−1(Ω)fulfills the estimates u(t) 2 L2(Ω) ≤M L+e−L t u(0) 2 L2(Ω) λAZt 0k∇uk2 L2(Ω) ds ≤1 2+β L u(0) 2 L2(Ω) +Cβ+βM Lt for every t∈[0, T]with the constants M:= 2 Cβ>0and L:= 2 λΩλA−2β > 0. The proof is given in §7. 4
Proposition 4 Suppose that conditions (a)–(c)hold for Fand that T < ∞. Then there exists positive constants C0,C1and C2depending only on β,Cβ, ΛA,λAand λΩsuch that every weak solution u∈L20, T ;H1 0(Ω)of (4) with u0∈L20, T;H−1(Ω)and ku0kL2(Ω) ≤ρsatisfies the a priori estimates ku(s)k2 L2(Ω) ≤C1+e−C0sρ2 ∇u(t) 2 L2(Ω) ≤C1·max{1,1 t}· 1+e−C2tρ2+ ΨC1·(1 + e−C2tρ2) for every 0≤s < t ≤T. The proof, given below, uses analogous estimates for the Galerkin approximations in the following lemma, which is proved in the §7. Lemma 5 Let {wk:k≥1}be an orthogonal basis of H1 0(Ω) as in Remark 1. For each n∈N, suppose that un(t) = n X k=1 unk(t)·wkis a solution of d dt (un(t), wk) + hAun(t), wki= (F(un(t)), wk) (un(0), wk) = (u0, wk), k = 1 . . . n. (11) If F:L2(Ω) →L2(Ω) satisfies the hypotheses (b)and (c), then there exist positive constants C1,C2and C3depending only on β,Cβ,ΛA,λAand λΩ such that whenever ku0kL2(Ω) ≤ρ, the following estimates holds for every s0,s, t∈(0, T]with 0< s0≤s≤t ∇un(t) 2 L2(Ω) ≤C1·max 1,1 t·1+e−C2tρ2+ ΨC1(1 + e−C2tρ2) Zt sku0 n(ξ)kL2dξ ≤C3·√t−s·const (s0, T, ρ). ku0 n(t)kH−1(Ω) ≤const (β, Cβ,ΛA, λA, λΩ, ρ). Proof of Proposition 4. The inclusions u∈L20, T;H1 0(Ω)and u0∈ L20, T;H−1(Ω)always imply that u∈C0[0, T ]; L2(Ω), see [7, §5.9]. Let {wk:k≥1}be an orthogonal basis of H1 0(Ω) as in Remark 1. Then for each n∈N, un: [0, T]→H1 0(Ω), t 7−→ n X k=1 u(t), wkwk is induced by the orthogonal projection of u(t) on span{w1. . . wn}in L2(Ω). It solves the perturbed nonlocal Galerkin problem (d dt (un(t), wk) + hAun(t), wki= (G(t), wk) (un(0), wk) = (u0, wk), k = 1 . . . n, (12) with the map G: [0, T]→L2(Ω) defined by G(t) := F(u(t)) for each t∈[0, T], which depends only on time in combination with the weak solution u(·) (but not 5
on unexplicitly). In particular, Gfulfills the conditions (a)–(c) (uniformly with respect to time). Hence, Lemma 5 provides a priori estimates for each Galerkin approximation un(·) depending essentially only on the L2norm of the initial value u0. Since {wk, k ≥1}is an orthonormal basis of L2(Ω), the sequence (un(t))n∈N converges to u(t) in L2(Ω) at each time t∈[0, T]. The L2bound of the H1 0 norm implies a weakly convergent subsequence of (un)n∈Nin L20, T;H1 0(Ω) and, its weak limit is u∈L20, T ;H1 0(Ω)(again). For each s0∈(0, T), we even have a L∞bound of the H1 0norms of (un)n∈Nin [s0, T] and so Lemma 7 below guarantees the same estimates holds for ∇u(t) 2 L2(Ω) at every time t∈ [s0, T], not just for Lebesgue-almost all such t. Fixing s0∈(0, T) arbitrarily, the Galerkin approximations of any weak solution u(·) with ku(0)kL2(Ω) ≤ρare equi–continuous w.r.t. L2(Ω) in [s0, T ] due to the second estimate in Lemma 5. Now the pointwise convergence of the Galerkin approximations to u(·) implies the following statement directly: Lemma 6 Suppose that the conditions (a)–(c)hold for Fand that T < ∞. For every ρ > 0, the subset of C0[0, T ]; L2(Ω)consisting of all weak solutions u∈L20, T;H1 0(Ω)of (4) with u0∈L20, T;H−1(Ω)and ku(0)kL2(Ω) ≤ρ is equi-continuous in the subinterval [s0, T]for every s0∈(0, T). Lemma 7 Let X, Y be Banach spaces such that Xis reflexive, and the inclusion X⊂Yis continuous. Assume that (un)n∈Nis a bounded sequence in L∞(t0, T;X)such that un* u weakly in Lp(t0, T;X)for some p∈[1,+∞) and u∈C0([t0, T]; Y). Then, for every t∈[t0, T ],u(t)belongs to Xand satisfies ku(t)kX≤sup n≥1kunkL∞(t0,T ;X). Proof of Lemma 7. We denote C:= sup n≥1kunkL∞(t0,T ;X). As (un)n∈Nis a bounded sequence in L∞(t0, T ;X), there exist a subsequence (uµ) and v∈L∞(t0, T;X) such that uµ ∗ * v in L∞(t0, T;X), i.e., ZT t0hw∗(t), uµ(t)idt −→ ZT t0hw∗(t), v(t)idt ∀w∗∈L1(t0, T ;X0), where by h·,·i we denote the duality product between X0and X. In particular, we have this convergence for all w∗∈Lp0(t0, T ;X0). Then, uµ* v weakly in Lp(t0, T;X), and as we also have un* u weakly in Lp(t0, T ;X), then v=u. 6
Then, uµ ∗ * u in L∞(t0, T;X) and by the ∗−weak lower semicontinuity of the norm, we obtain kukL∞(t0,T ;X)≤lim inf µ→∞ kuµkL∞(t0,T ;X)≤C. (13) Now fix t∈[t0, T]. By (13) there exists a sequence (tn)n∈Nin [t0, T ] such that tn→tand u(tn)∈Xwith ku(tn)kX≤Cfor all n∈N. As Xis reflexive, there exist a subsequence (tµ) and x∈Xsuch that u(tµ)* x weakly in X. The inclusion X⊂Yis assumed to be continuous and so, u(tµ)* x weakly in Y. (14) Due to u∈C0([t0, T]; Y), we have in addition that u(tn)→u(t) in Y. This implies u(t) = x∈X. Finally the weak lower semi-continuity of the norm implies for every t∈[t0, T ] ku(t)kX=kxkX≤lim inf µ→∞ ku(tµ)kX≤C 4 Existence of weak solutions Proposition 8 Suppose that hypotheses (a)–(c)hold for F:L2(Ω) →L2(Ω). Then, for every u0∈L2(Ω), there exists a weak solution u∈L20, T;H1 0(Ω) of problem (4). Moreover, ubelongs to C0[0, T]; L2(Ω)and, u|(0,T ]is locally bounded in H1 0(Ω). Furthermore, if Fis locally Lipschitz as in hypothesis (d), then the weak solution is unique. Proof. The proof is based on a sequence of Galerkin approximations and the a priori estimates in §3. Let {wk, k ≥1}be an orthogonal basis of H1 0(Ω) as in Remark 1 and for each n∈N, consider a solution un(t) = n X k=1 unk(t)wkof (11). Fix s0∈(0, T) arbitrarily. By Proposition 3, the sequence (un)n∈Nis bounded in L20, T ;H1 0(Ω)Due to Lemma 5, (u0 n)n∈Nis bounded both in L2s0, T ;L2(Ω)and L2(0, T ;H−1(Ω)). Hence Alaoglu’s Theorem provides a subsequence (again denoted by) (un)n∈N and functions u∈L20, T;H1 0(Ω),v∈L2s0, T;L2(Ω),w∈L20, T;H−1(Ω) with un−→ uweakly in L20, T;H1 0(Ω) u0 n−→ vweakly in L2s0, T;L2(Ω) u0 n−→ wweakly in L20, T;H−1(Ω). 7
In particular, u0=v=wholds Lebesgue-almost everywhere, which a simple check of the distributional derivative property reveals. This implies that u∈ L2(0, T;H−1(Ω)) ∩C0[0, T]; L2(Ω). Standard arguments conclude u(0) = u0 from un(0) −→ u0in L2(Ω) for n→ ∞ (see e.g. [7, §7.1]). The sequence (un)n∈Nis equi–continuous in C0[s0, T ]; L2(Ω)by Lemma 6. Moreover, the set un(t)t∈[s0, T], n ∈Nis relatively compact in L2(Ω) as a consequence of Lemma 5 and the Sobolev Embedding Theorem. Hence the Arzel`a–Ascoli Theorem provides a further subsequence (again denoted by) (un)n∈Nwith un(t)−→ u(t) in L2(Ω) uniformly for t∈[s0, T]. Hypothesis (a) on the continuity of Fimplies that F(un(t)) → F(u(t)) in L2(Ω) for every t∈[s0, T]. Taking the limit as n→∞gives that u∈L2s0, T;H1 0(Ω)is a weak solution to the partial differential equation in problem (4) with u0∈L2s0, T;H−1(Ω). Finally, u|[s0,T ]: [s0, T ]→H1 0(Ω) is bounded due to Proposition 4. If Fsatisfies the local Lipschitz condition (d) in addition the uniqueness of the weak solution follows by a standard argument. 5 The multivalued semiflow of weak solutions Define ΦF: [0,∞)×L2(Ω) → PL2(Ω), where PL2(Ω)consists of all nonempty subsets of L2(Ω), by ΦF(t, u0) := u(t)∈L2(Ω) ∃u(·)∈L20, t;H1 0(Ω): u0∈L20, t;H−1(Ω), u(0) = u0and uis weak solution of (4) in Ω ×(0, t). (15) It is clear that this multivalued map ΦFforms a strict multivalued semiflow on L2(Ω) as in the the following definition of Kapustyan et al. [9, Definition 2.1]. Definition 9 Let (X, d)be a metric space and, P(X)consists of all its nonempty subsets. A map Φ : [0,∞)×X→ P(X)is called strict multivalued semiflow (m–semiflow) on Xif it satisfies the following conditions: (A) Φ(0, x) = {x}for all x∈X (B) Φ(t+s, x) = Φ(t, Φ(s, x)) for all s, t ≥0, x ∈X. Remark 10 Kapustyan et al. [9, Definition 2.1] define, in fact, a more general m–semiflow Φ : [0,∞)×X→ P(X), which satisfies condition (A), but 8
instead of the equality condition (B), satisfies the following weaker inclusion condition (C) Φ(t+s, x)⊂Φ(t, Φ(s, x)) for all s, t ≥0, x ∈X. The reason is that the counterpart of the setvalued mapping ΦFabove for many systems, in particular the 3-dimensional Navier–Stokes equations, is restricted to weak solutions that satisfy an energy inequality. However, such energy inequalities only hold (or can only be proved to hold) for almost all time instants. Hence a concatentation of weak solutions satisfying the energy inequality on adjacent time intervals may not satisfy the energy inequality on the concatenated time interval, which means that the strict property (B) need not hold. see also Morillas & Valero [12]. This situation does not arise for the mapping ΦFdefined above for the problem (4) under consideration in this paper. Every strict multivalued semiflow Φ : [0,∞)×X→ P(X) (in the sense of Definition 9) induces a map ˜ Φ : [0,∞)× P(X)→ P(X) in the following canonical way: ˜ Φ(t, M)Def. =[ x∈M Φ(t, x) for all nonempty M⊂X. Obviously, it satisfies the corresponding conditions (A∗)˜ Φ(0, M) = Mfor all M∈ P(X) (B∗)˜ Φ(t+s, M) = ˜ Φ(t, ˜ Φ(s, M)) for all s, t ≥0, M ∈ P(X) or in the general nonstrict case (C∗)˜ Φ(t+s, M)⊂˜ Φ(t, ˜ Φ(s, M)) for all s, t ≥0, M ∈ P(X) Hence one usually does not distinguish between ˜ Φ and Φ. 5.1 Global attractors of multivalued semiflows: General results Definition 11 Aglobal attractor of a strict multivalued semiflow Φon a metric space (X, d)is a nonempty subset Aof Xwhich is Φ-invariant, i.e. Φ(t, A) = Afor all t≥0, and attracts every bounded subset Bof X, i.e. distX(Φ(t, B), A)→0as t→ ∞, where distXis the Hausdorrf semi–distance on P(X). For a general non-strict multivalued semiflow the attractor Ais only required to be Φ-negativly invariant, i.e. A⊂Φ(t, A)for all t≥0 [11, Definition 6]. The existence of a global attractor is usually concluded from the compactness or asymptotic compactness of the multivalued mapping Φ(t, ·) for t > 0: 9
we observe that if we consider m≥1, we have ZΩ|u|mdx ≤|Ω| |B(0; δ)|m/2kukm L2(Ω) . (18) We also obtain ZΩ (F(u))2dx ≤C|Ω| |B(0; δ)|kuk2 L2(Ω) +C|Ω| |B(0; δ)|2kuk4 L2(Ω) +2C|Ω| |B(0; δ)|3/2kuk3 L2(Ω) , and then we have (7) with ΦkukL2(Ω)=C|Ω| |B(0;δ)|kuk2 L2(Ω) +C|Ω| |B(0;δ)|2kuk4 L2(Ω) + 2C|Ω| |B(0;δ)|3/2kuk3 L2(Ω). We observe that if u∈L2(Ω), then F(u)∈L∞(Ω) ⊂L2(Ω) . Now, we prove that Fis continuous. We suppose that un−→ ustrongly in L2(Ω), then we have to prove that F(un)−→ F(u) strongly in L2(Ω). It is sufficient to prove that there exists {uµ}⊂{un}such that F(uµ)−→ F(u) strongly in L2(Ω). As un−→ ustrongly in L2(Ω), then (see [4]) there exists {uµ}⊂{un}such that uµ(x)−→ u(x) a.e. in Ω. Then, as fis continuous we have f(uµ(x)) −→ f(u(x)) a.e. in Ω. (19) We observe that arguing as before we obtain |uµ(x)−u(x)|=1 |B(x;δ)|ZB(x;δ)∩Ω (uµ(y)−u(y)) dy ≤1 |B(0; δ)|1/2kuµ−ukL2(Ω) −→ 0, and then uµ(x)−→ u(x) a.e. in Ω. (20) From (19) and (20) we have F(uµ)(x)−→ F(u)(x) a.e. in Ω. (21) 16
On the other hand, we observe that kuµkL2(Ω) is bounded because Ω is a bounded domain and uµ−→ ustrongly in L2(Ω). Then, we have |F(uµ(x))| ≤ C|uµ(x)||1−uµ(x)|(22) ≤C1 |B(0; δ)|1/2kuµkL2(Ω) 1 + 1 |B(0; δ)|1/2kuµkL2(Ω)! ≤e C, where e Cis a positive constant. Then, from (21), (22) and by the Dominated Convergence Theorem, we obatin F(uµ)−→ F(u) strongly in L2(Ω) , then Fis continuous. From Theorem 8 we have that there exists a weak solution u∈L20, T ;H1 0(Ω) of problem (16) with u0∈L20, T;H−1(Ω).According to Theorem 15, the problem (16) defines a multivalued semiflow in L2(Ω), which possesses a compact global attractor A. Also, Ais the minimal closed attracting set. 7 Proofs of the dissipativity estimates Proof of Proposition 3. The uniform ellipticity condition of the linear operator Aand assumption (b) on the nonlocal operator Fused in the energy equality, 1 2 d dt kuk2 L2(Ω) +ZΩ u(A u)dx =ZΩ uF(u)dx (23) give 1 2 d dt kuk2 L2(Ω) +λAk∇uk2 L2(Ω) ≤βkuk2 L2(Ω) +Cβ.(24) Hence, by the Poincar´e inequality (3), 1 2 d dt kuk2 L2(Ω) +λΩλAkuk2 L2(Ω) ≤βkuk2 L2(Ω) +Cβ. i.e., d dt kuk2 L2(Ω) ≤M−Lkuk2 L2(Ω) , (25) where M:= 2 Cβ>0 and L:= 2 λΩλA−2β > 0. Multiplying by eLt, and integrating between 0 and t, it follows eLt ku(t)k2 L2(Ω) ≤ ku(0)k2 L2(Ω) +MZt 0 eLs ds ≤ ku(0)k2 L2(Ω) +MZt −∞ eLs ds, 17
i.e., for every t∈[0, T], ku(t)k2 L2(Ω) ≤M L+e−Lt ku(0)k2 L2(Ω) . (26) Finally, integrating equation (24) with respect to time, we obtain λAZt 0k∇uk2 L2(Ω) ds ≤1 2ku0k2 L2(Ω) −ku(t)k2 L2(Ω)+βZt 0kuk2 L2(Ω) ds +Cβt ≤ ku0k2 L2(Ω) 1 2+βZt 0 e−L s ds+Cβ+βM Lt. Proof of Lemma 5. Let {wk:k≥1}denote an orthogonal basis of H1 0(Ω) as mentioned in Remark 1. For each n≥1, let un(t) = n X k=1 unk(t)wk, be a Galerkin approximation of problem (4) satisfying the finite dimensional system of ordinary differential equations in the sense that (d dt (un(t), wk) + hAun(t), wki= (F(un(t)), wk) (un(0), wk)=(u0, wk), k = 1, . . . , n. (27) We observe that Aun(t) = n X k=1 unk(t)Awk= n X k=1 unk(t)λkwk∈span {w1, . . . , wn}. Then, multiplying equation (27) by unk(t)λk, summing from k= 1 to n, and using property (c) of F, gives hu0 n(t), Aun(t)i+kAun(t)k2 L2(Ω) = (F(un(t)), Aun(t)) ≤1 2kF(un(t))k2 L2(Ω) +kAun(t)k2 L2(Ω) ≤Ψ(un(t)) + 1 2kAun(t)k2 L2(Ω) (28) 18
for Lebesgue-almost all t∈[0, T ]. On the other hand, hu0 n(t), Aun(t)i=−ZΩ N X i,j=1 ∂xjaij(x)∂xiunu0 ndx (29) = N X i,j=1 ZΩ aij(x)∂xiun(∂t∂xjun)dx =d dt 1 2·A[un, un](30) for the symmetric bilinear form A[u, v] := ZΩ N X i,j=1 aij(x)∂xiu ∂xjv dx,u, v ∈H1 0(Ω). since it is assumed that aij =aji, with i, j = 1, .., N, and these coefficients do not depend on t. Integrating from sand t, then gives 1 2·A[un(t), un(t)] ≤1 2·A[un(s), un(s)] + Zt s Ψkun(ξ)kL2(Ω)dξ. (31) Hence, it follows from the inequalities of coercivity and continuity λAk∇un(s)k2 L2(Ω) ≤ A[un(s), un(s)] ≤ΛAk∇un(s)k2 L2(Ω) that λA 2·k∇un(t)k2 L2(Ω) ≤ΛA 2·k∇un(s)k2 L2(Ω) +Zt s Ψkun(ξ)kL2(Ω)dξ. (32) Integrating now the variable sbetween t0:= max{0, t −1}and t, gives λA 2·min{t, 1}·k∇un(t)k2 L2(Ω) ≤ΛA 2Zt max{0,t−1}k∇un(s)k2 L2(Ω) ds + Zt max{0,t−1} Ψkun(s)kL2(Ω)ds. On the other hand, integrating (24) with respect to time in [t0, t], it follows that kun(t)k2 L2(Ω) −kun(t0)k2 L2(Ω) + 2 λAZt t0k∇un(s)k2 L2(Ω) ds ≤2βZt t0kun(s)k2 L2(Ω) ds + 2 Cβ. 19
Then, using Proposition 3, ΛAZt t0k∇un(s)k2 L2(Ω) ds ≤ΛA 2β+ 1 2λA sup s∈[t0,t]kun(s)k2 L2(Ω) +CβΛA λA ≤C2·1 + e−L t ku0k2 L2(Ω)(33) for all t≥0 with a constant C2=C2(β, ΛA, λA, λΩ, Cβ)>1. Thus, whenever ku0kL2(Ω) ≤ρ, k∇un(t)k2 L2(Ω) ≤2C2 λAmin{t, 1}1 + e−L t ρ2+ ΨM L+e−L t ρ2.(34) Moreover, this upper bound holds for every t∈(0, T] — in contrast to the immediate consequence of inequality (28) and the coercivity of A. The next step is to establish an estimate for ku0 nkL∞(0,T ;H−1(Ω)). This follows essentially the arguments for the standard energy estimate, taking the preceding a priori bounds into consideration. For every v∈H1 0(Ω), there is a unique representation v=v1+v2with v1∈ span{w1, . . . , wn}and (v2, wk) = 0 for k= 1, . . .,nsince {wk, k ≥1}is an orthonormal basis of L2(Ω). For Lebesgue-almost every t∈[0, T ] it follows that u0 n(t), v1+Aun(t), v1=F(un(t)), v1 hu0 n(t), vi= (u0 n(t), v) = u0 n(t), v1 =F(un(t)), v1−Aun(t), v1 ≤(kF(un(t))kL2+ ΛAk∇un(t)kL2)kv1kH1 0(Ω). Then, inequality (7) in assumption (c) of Fimplies that hu0 n(t), vi ≤ pΨ (kun(t)kL2)+ΛAk∇un(t)kL2kv1kH1 0(Ω). Hence, in view of the estimates (26) and (34), ku0 n(t)kH−1(Ω) ≤pΨ (kun(t)kL2)+ΛAk∇un(t)kL2 ≤const β, Cβ,ΛA, λA, λΩ,ku0kL2(Ω). Finally, multiplying (27) by u0 nk(t) and summing from k= 1 to n, it follows 20
from inequality (10) in assumption (c) of Fthat hu0 n(t), u0 n(t)i+hAun(t), u0 n(t)i= (F(un(t)), u0 n(t)) ≤ kF(un(t))k2 L2(Ω) +1 4ku0 n(t)k2 L2(Ω) ≤Ψkun(t)kL2(Ω)+1 4ku0 n(t)k2 L2(Ω). Using (30), this can be reformulated as ku0 n(t)k2 L2(Ω) +d dt 1 2A[un, un]≤Ψkun(t)kL2(Ω)+1 4ku0 n(t)k2 L2(Ω) i.e., for Lebesgue-almost every t∈[0, T ], ku0 n(t)k2 L2(Ω) +d dt A[un, un]≤2 Ψkun(t)kL2(Ω). Now fix s0∈(0, T[ arbitrarily and integrate between s0and Tto obtain ZT s0ku0 n(ξ)k2 L2(Ω) dξ +A[un(T), un(T)] ≤A[un(s0), un(s0)] + 2 ZT s0 Ψkun(ξ)kL2(Ω)dξ. The general inequalities of coercivity and continuity 0≤λAk∇un(t)k2 L2(Ω) ≤ A[un(t), un(t)] ≤ΛAk∇un(t)k2 L2(Ω) , imply that ZT s0ku0 n(ξ)k2 L2(Ω) dξ ≤ΛAk∇un(s0)k2 L2(Ω) + 2 ZT s0 Ψkun(ξ)kL2(Ω)dξ. Hence, from the estimates (26), (34) and the monotonicity of Ψ one concludes that ZT s0ku0 n(ξ)k2 L2(Ω) dξ ≤const(β, ΛA, λA, λΩ, Cβ, s0, T, ku0kL2(Ω)). Finally, H¨older’s inequality guarantees the claimed inequality for every s, t ∈ [s0, T] with s<t, i.e., Zt sku0 n(ξ)kL2(Ω) dξ ≤const β, ΛA, λA, λΩ, Cβ, s0, T, ku0kL2(Ω)·√t−s . 21
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