The dimension of attractors of nonautonomous partial differential equations

Dimension of attractors of nonautonomous partial differential equations T. Caraballo∗J.A. Langa†J. Valero‡ Abstract The concept of nonautonomous (or cocycle) attractor has become a proper tool for the study of the asymptotic behaviour of general nonautonomous partial differential equations. This is a time-dependent family of compact sets, invariant for the associated process and attracting “from −∞”.In general, the concept is rather different from the classical one of global attractor for autonomous dynamical systems. We prove a general result on the finite fractal dimensionality of each compact set of this family. In this way, we generalize previous results of Chepyzhov and Vishik in [6]. Our results are also applied to differential equations with a nonlinear term having polynomial growth at most. Contents 1 Introduction 2 2 Attractors of nonautonomous equations 3 3 Dimension of nonautonomous attractors 5 4 Applications to a nonautonomous partial differential equation 10 ∗Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. E-mail: [email protected] †Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. E-mail: [email protected] ‡Universidad Cardenal Herrera CEU, Comisario 3, 03203 Elche, Alicante, Spain. Email: v[email protected] 1 1 Introduction In this paper, we develop a general theory on the finite dimension of attractors for nonautonomous partial differential equations and we apply it, in particular, to estimate the fractal dimension of the attractor for the following nonautonomous equation      ∂u ∂t −∆u+f(t, u) = h(t), u|∂Ω= 0, u(τ) = uτ, where the function h(t) is allowed to have polynomial growth in time (see condition (9) below). For these kind of nonautonomous systems it is not possible in general to obtain a uniform global attractor in the sense of [5], since the trajectories can be unbounded when time rises to infinity. A different approach was developed in [8], [9], [25] (see also [2], [17], [16], [24]), where the existence of attractors for some stochastic and nonautonomous equations is studied. The main definitions and theorems from the abstract theory of attractors for such systems are given in Section 2. It is worth pointing out that in such systems the global attractor is not a compact set, but a parameterized family A(t) of compact sets. We are interested in proving the finite dimensionality of each of the sets A(t). We note that the union of all the attractors, i.e. ∪t∈ R A(t) can be infinite-dimensional. In the case of stochastic equations of parabolic and hyperbolic types such results were obtained in [10], [12], [13]. There are some technical tools in the proofs of these papers that do not seem to be applicable to the nonautonomous case. As far as we know, the only result in the nonautonomous case, was proved in [4] under the assumption of being the function h(t) uniformly bounded in the variable t. In such case, the union of the whole family of attractors ∪t∈ R A(t) is bounded, and the well known technique of Lyapunov exponents, developed in [7], can be adapted with slight modifications. However, when the function h(t) is allowed to have polynomial growth, the supremum of the norm of the global attractor A(t) can have also polynomial growth, so that we cannot expect that the union of attractors is bounded. In this paper we extend the general theory on the finite-dimensionality of compact invariant sets in Hilbert spaces (see [1], [15], [20], [23], [27]) to the case of a parameterized family of global attractors with polynomial growth at most. The invariance property for nonautonomous attractors is now stated for a time-dependent family of compact sets {A(t)}t∈ R and the attraction is defined for trajectories with initial time going to −∞.Thus, the idea is to construct a sequence of coverings of A(t) by iterating ntimes an initial 2 covering of A(t−nT ∗),as n→ ∞. Further we apply this abstract theorem to the attractor of the equation given above. We note that we are able to obtain the estimation of dimension in the case where the function f(t, u) is globally Lipschitz on the second variable u. In the autonomous case it is possible to change the global Lipschitz condition by a local one by proving that the global attractor is bounded in L∞(Ω) (see [14], [21], [28]). In our case, in order to use a similar idea we would need to obtain an estimation of the norm in L∞(Ω) of the union ∪τ≤tA(τ),∀t, which is not possible in general as we have already remarked. 2 Attractors of nonautonomous equations In this section, we introduce the general framework in which the theory of attractors for nonautonomous systems is going to be studied (see Crauel et al. [9] and Schmalfuss [26]). In a first step, we define semiprocesses as two-time dependent operators related with the solutions of nonautonomous differential equations. In this way, we are able to treat these equations as dynamical systems. Secondly, we write the general definitions of invariance, absorption and attraction and we finish with a general theorem on the existence of global attractors for these equations. Let (H, d) be a complete metric space (with the metric d) and {S(t, s)}t≥s, t, s ∈Rbe a family of mappings satisfying: i) S(t, t, ·) = Id, ii) S(t, s, S(s, τ, u)) = S(t, τ, u),for all τ≤s≤t, u ∈H, iii) u7→ S(t, τ, u) is continuous in H. This map is called a process (this term was introduce by Dafermos [11]). In general, we have to consider S(t, τ, u) as the solution of a nonautonomous equation at time twith initial condition uat time τ. Let Dbe a non-empty set of parameterized families of non-empty bounded sets b D={D(t)}t∈ R . In particular, b D={D(t)}t∈ R ∈ D, where D(t)≡B for all t, and B⊂His a bounded set. In what follows, we will consider fixed this set D,so that the concepts of absorption and attraction in our analysis are always referred to it. For A, B ⊂Hwe define the Hausdorff semidistance as, dist(A, B) = sup a∈A inf b∈Bd(a, b). 3 Definition 2.1 Given t0∈R, we say that K(t)⊂His attracting at time t0 if for every b D={D(t)} ∈ D we have that lim τ→−∞dist(S(t0, τ, D (τ)), K(t0)) = 0. A family b K={K(t)}t∈ R is attracting if K(t0)is attracting at time t0,for all t0∈R. The previous concept considers a fixed final time and moves the initial time to −∞. Note that this does not mean that we are going backwards in time, but we consider the state of the system at time t0starting at τ→ −∞. This is called pullback attraction in the literature (cf. [18], [26]). Definition 2.2 Given t0∈R, we say that B(t0)⊂His absorbing at time t0if for every b D={D(t)} ∈ D there exists T=T(t, b D)∈Rsuch that S(t0, τ, D (τ)) ⊂B(t0),for all τ≤T. A family b B={B(t)}t∈ R is absorbing if B(t0)is absorbing at time t0,for all t0∈R. Note that every absorbing set at time t0is attracting. Definition 2.3 Let b B={B(t)}t∈ R be a family of subsets of H. This family is said to be invariant with respect to the process Sif S(t, τ, B(τ)) = B(t),for all (τ, t)∈R2, τ ≤t. Note that this property is a generalization of the classical property of invariance for semigroups. However, in this case we have to define the invariance with respect to a family of sets depending on a parameter. We define the omega-limit set at time t0of b D≡ {D(t)} ∈ D as Λ( b D, t0) = \ s≤t0[ τ≤s S(t0, τ, D (τ)). From now on, we assume that there exists a family b K={K(t)}t∈ R of compact absorbing sets, that is, K(t)⊂His non-empty, compact and absorbing for each t∈R. Note that, in this case, Λ( b D, t0)⊂K(t0),for all b D={D(t)} ∈ D,t0∈R.As in the autonomous case, it is not difficult to prove that under these conditions Λ( b D, t0) is non-empty, compact and attracts b D={D(t)} ∈ D at time t0. The proof is similar to that of [9, Lemma 1.1], where the set Dconsists only of bounded sets. 4 Definition 2.4 The family of compact sets b A={A(t)}t∈ R is said to be the global attractor associated to the process Sif it is invariant, attracting every b D={D(t)} ∈ D (for all t0∈R) and minimal in the sense that if b C={C(t)}t∈ R is another family of closed attracting sets, then A(t)⊂C(t) for all t∈R. Remark 2.5 Chepyzhov and Vishik [4] define the concept of kernel sections for nonautonomous dynamical systems which corresponds to our definition of global nonautonomous attractor with b D={D(t)≡B}t∈ R where B⊂H is bounded. The general result on the existence of nonautonomous attractors is a generalization of the abstract theory for autonomous dynamical systems (Temam [27], Hale [19]): Theorem 2.6 Assume that there exists a family of compact absorbing sets. Then, the family b A={A(t)}t∈ R defined by A(t) = [ b D∈D Λ( b D, t) is the global nonautonomous attractor. As the proof of this theorem repeats the same one of [9, Theorem 1.1] with slight modifications, we will omit it. Remark 2.7 All the general theory of nonautonomous attractors can be written in the framework of cocycles (cf., among others, Cheban et al. [2], Crauel and Flandoli [8], Kloeden and Schmalfuss [18], Schmalfuss [26]). We could have also followed this notation here, but we think that, in this case, it is more clear to keep the explicit dependence on time of the attractor, which in addition, allows us to compare more straightforward the results in [4]. 3 Dimension of nonautonomous attractors In [4], Chepyzhov and Vishik prove a general result for the Hausdorff dimension of kernel sections A(t) associated to a process {S(t, τ)}generated by a nonautonomous differential equation. The main hypothesis is the uniform boundedness of the set ∪t∈ R A(t). In applications, this is related to the existence of a uniform bound for the nonautonomous terms in the system. In our case, we allow these terms to be unbounded in t, so that their results are 5 not suitable for our situation. However, we are able to prove a general result on the finite fractal dimensionality of the nonautonomous attractor. Due to the weaker properties on the nonautonomous terms, it is not expected that a uniform bound for all tis obtained. Let Hbe a Hilbert space and A ⊂ Hbe a compact subset of H. We firstly recall the definition of the Hausdorff and fractal dimensions of A. We shall denote by B(a, r) a closed ball of radius rcentered at a. Let U be a covering of Aby a finite family of balls B(xi, ri) such that supi(ri) = δ(U)≤δ. Then the d-dimensional Hausdorff measure of Ais defined as follows: µH(A, d) = lim δ→0µH(A, d, δ), where µH(A, d, δ) = inf δ(U)≤δX i rd i, where the inf is extended to all the possible covering Uof Asuch that δ(U)≤ δ. It is known that there exists d=dH(A)∈[0,+∞] such that µH(A, d) = 0 for d > dH(A) and µH(A, d) = ∞for d < dH(A). The value dH(A) is called the Hausdorff dimension of A. The fractal dimension of Ais given by df(A) = inf{d > 0|µf(A, d) = 0}, where µf(A, d) = lim ²→0µf(A, ε, d) = lim ²→0²dn², and n²is the minimum number of balls of radius r=²which is necessary to cover A. Since µH(A, d)≤µf(A, d) it is clear that dH(A)≤df(A), the converse being false in general (Eden et al. [15]). Before proving our main result in this section, we will recall a technical lemma which will be repeatedly used in the proof (see Lemma 1 in [1]). Lemma 3.1 Let B(a, γ)⊂RNbe a closed ball centered at aof radius γ. For any 0<λ<γthe minimum number of balls nλof radius λwhich is necessary to cover B(a, γ)is less or equal to ¡3γ λ¢N. We consider now a process S(t, τ, u) : R×R×H→H, t ≥τ, having the family of global attractors b A={A(t)}t∈ R . Theorem 3.2 Suppose there exist constants K0, K1, θ > 0such that kA(t)k+≤K0|t|θ+K1,∀t∈R,(1) 6 where kA(t)k+= supy∈A(t)kyk. Also assume that for any t∈Rthere exist T∗=T∗(t),l=l(t, T ∗)∈ [1,+∞), δ =δ(t, T ∗)∈(0,1 √2)and N=N(t),such that for any u, v ∈ A(τ), τ ≤t−T∗, kS(τ+T∗, τ, u)−S(τ+T∗, τ, v)k ≤ lku−vk,(2) kQN(S(τ+T∗, τ, u)−S(τ+T∗, τ, v))k ≤ δku−vk,(3) where QNis the projector mapping Honto some subspace H⊥ Nof codimension N∈N. Then, for any η=η(t)>0such that σ=σ(t) = ¡6√2l¢N¡√2δ¢η< 1,the next inequality holds dH(A(t)) ≤df(A(t)) ≤N+η. (4) Proof. Let us fix t∈Rand choose η > 0 such that σ < 1. We also take an arbitrary τ≤t−T∗,and denote ε(τ) = 2 ³K0|τ|θ+K1´. Let U0be a covering of A(τ) by one ball B(a1, ε (τ)), a1∈A(τ), of radius ε(τ) centered at a1. Hence, A(τ)⊂B(a1, ε (τ)). Since A(τ+T∗) = S(τ+T∗, τ, A(τ)) and using condition (2) we have A(τ+T∗)⊂B(S(τ+T∗, τ, a1), lε (τ)). Let us denote by PNthe orthoprojector onto the subspace HNof dimension Nwhich is orthogonal to H⊥ N(and then PN⊕QN=I,HN⊕H⊥ N=H). It is clear that PNB(S(τ+T∗, τ, a1), lε (τ)) ⊂BN(PNS(τ+T∗, τ, a1), lε (τ)), where BN(a, β) denotes a closed ball in HNof radius βand centered at a. In view of the preceding lemma we can cover BN(PNS(τ+T∗, τ, a1), lε (τ)) by balls BN(a1j,δ 2ε(τ)), j= 1, ..., m1,a1j∈HNand m1=m1(t)≤µ6l δ¶N . Let us denote M1j= (P−1 NBN(a1j,δ 2²(τ))) ∩ A(τ+T∗). We take arbitrary y1j∈ M1j. We shall show that the set of balls B(y1j, γ² (τ)), j= 1, ..., m1,γ=√2δ(note that we have assumed that γ < 1), is a new covering of A(τ+T∗). Since A(τ+T∗)⊂ m1 [ j=1 M1j, it is sufficient to prove that M1j⊂B(y1j, γ² (τ)),∀j. Let y∈ M1j. There exist v1, v2∈B(a1, ε (τ))∩A(τ) such that S(τ+T∗, τ, v1) = y, S(τ+T∗, τ, v2) = 7 y1j. Then kv1−v2k ≤ ²(τ) and in view of (3), kQNy−QNy1jk ≤ δ² (τ). On the other hand, kPNy−PNy1jk ≤ kPNy−a1jk+kPNy1j−a1jk ≤ δ² (τ). Hence, ky−y1jk ≤ q(δ² (τ))2+ (δ² (τ))2=γ² (τ). We have obtained a covering U1of A(τ+T∗) by balls of radius γ² (τ) such that the number of balls is m1. Therefore, nγ²(τ)≤m1≤µ6l δ¶N , where nγε(τ)denotes now the minimum number of balls of radius equal to γε (τ) which is necessary to cover A(τ+T∗). Then, µf(A(τ+T∗), γ² (τ), d) = nγ²(τ)(γ² (τ))d≤µ6l δ¶N³√2δ² (τ)´d . Taking d=d(t) = N+η, we get µf(A(τ+T∗), γ² (τ), N +η)≤³√2δ´η³6√2l´N ²(τ)N+η=σ² (τ)N+η. Suppose now that τ≤t−2T∗. Take the covering U1={B(y1i, γε (τ))}m1 i=1 of A(τ+T∗) and define Mi=S(τ+ 2T∗, τ +T∗, A (τ+T∗)∩B(y1i, γε (τ)))∩ A(τ+ 2T∗), i = 1, ..., m1. Now, since A(τ+ 2T∗) = S(τ+ 2T∗, τ +T∗,A(τ+T∗))) and using condition (2), we have A(τ+ 2T∗)⊂ m1 [ i=1 Mi⊂ m1 [ i=1 B(S(τ+ 2T∗, τ +T∗, y1i), lγε (τ)). It is clear that PNB(S(τ+ 2T∗, τ +T∗, y1i), lγε (τ)) ⊂BN(PNS(τ+ 2T∗, τ +T∗, y1i), lγε (τ)),∀i. In view of the preceding technical lemma, we can cover each BN(PNS(τ+ 2T∗, τ +T∗, y1i), lγε (τ)) by balls BN(aij,δ 2γε (τ)), j= 1, ..., ni,aij ∈HNand ni=ni(t)≤µ6l δ¶N ,∀i. Let us denote Mij = (P−1 NBN(aij,δ 2γ² (τ))) ∩ Mi. We take arbitrary yij ∈ Mij. We shall show that the set of balls B(yij, γ2²), i= 1, ..., m1, j = 1, ..., ni,γ=√2δ, is a new covering of A(τ+ 2T∗). Indeed, since A(τ+ 2T∗)⊂[ ij Mij, 8 it is sufficient to prove that Mij ⊂B(yij, γ2²),∀i, j. Let y∈ Mij. There exist v1, v2∈B(y1i, γε (τ)) ∩A(τ+T∗) such that S(τ+ 2T∗, τ +T∗, v1) = y, S(τ+ 2T∗, τ +T∗, v2) = yij. Then kv1−v2k ≤ γ² (τ) and in view of (3), kQNy−QNyijk ≤ δγ² (τ). On the other hand, kPNy−PNyijk ≤ kPNy−aijk+kPNyij −aijk ≤ δγ² (τ). Hence, ky−yijk ≤ q(δγ²)2+ (δγ²)2=γ2²(τ). We have obtained a covering U2of A(τ+ 2T∗) by balls of radius γ2²such that the number of balls is m2=m2(t) = Pm1 i=1 ni. Therefore, nγ2²≤ m1 X i=1 ni≤m1µ6l δ¶N ≤µ6l δ¶2N . Let k∈N. If we suppose that τ≤t−kT ∗,we can obtain, in the same way as before, a sequence of coverings Uj, j = 1,2, ..., k of the sets A(τ+jT ∗) by balls of radius γj²and such that the number of balls is less than or equal to ¡6l δ¢jN . Therefore, nγj²≤µ6l δ¶jN , where nγjε(τ)denotes now the minimum number of balls of radius equal to γjε(τ) which is necessary to cover A(τ+jT ∗). Hence, choosing τ=t−kT ∗we obtain µf(A(t), γk²(τ), N +η)≤³√2δ´kη ³6√2l´kN ε(τ)N+η ≤σk³K1+K0|(t−kT ∗)|θ´N+η . This implies that limα→0µf(A(t), α, d) = 0,for d=N+η. Indeed, as for k large enough the sequence r(k) = γkε(t−kT ∗) = γk³K1+K0|t−kT ∗|θ´ is decreasing, we have that for any α > 0 small enough, one can find some k∈Nsuch that r(k)≤α < r (k−1). It is clear that nα≤nr(k)≤¡6l δ¢kN . Then lim α→0µf(A(t), α, N +η) = lim α→0nααN+η≤lim k→∞ µ6l δ¶kN (r(k−1))N+η = lim k→∞ σkÃK1+K0|t−(k−1) T∗|θ γ!N+η = 0. Hence, dH(A(t)) ≤df(A(t)) ≤N+η. 9 where α > 0. It follows from (20) that T∗=log 2γ2 2(λN+1−c5). Hence, (21) will be satisfied if the next inequality holds (λN+1 −c5) log µc5+λN+1 (2γ2+α)ξ(t))¶≥4c5log √2γ. Using the inequality λN+1 ≥D(N+ 1)2 nwe get (λN+1 −c5) log µc5+λN+1 (2γ2+α)ξ(t)¶≥³(N+ 1)2 nD−c5´log Ãc5+D(N+ 1)2 n (2γ2+α)ξ(t)!. Choosing N=N(t) such that D(N+ 1)2 n≥5c5and c5+D(N+1) 2 n (2γ2+α)ξ(t)≥√2γthe inequality (21) holds. Hence, it is sufficient to choose Nsatisfying N≥max ((D1ξ(t))n 2−1,µ5c5 D¶n 2 −1), where D1=√2γ(2γ2+α) D. We take N= max n[(D1ξ(t))n 2],h¡5c5 D¢n 2io, where [x] denotes the integer part of x, and then N≤max n(D1ξ(t))n 2,¡5c5 D¢n 2o. Finally, Theorem 3.2 implies that df(A(t)) ≤2N≤2 max ((D1ξ(t))n 2,µ5c5 D¶n 2) = max nL1(ξ(t))n 2, L2(c5)n 2o, where L1= 2 (D1)n 2, L2= 2 ¡5 D¢n 2. From the previous result we can also obtain a uniform bound in tfor the fractal dimension of the attractors: Corollary 4.7 There exists a positive constant Kdepending on n, Ω, c5and ξ(·)(but not on t) such that dH(A(t)) ≤df(A(t)) ≤K, for all t∈R. Proof. Fix some t∗∈R.Since ξ(t) is non-decreasing, Theorem 4.6 gives df(A(t)) ≤max nL1(ξ(t))n 2, L2(c5)n 2o ≤max nL1(ξ(t∗))n 2, L2(c5)n 2o, for all t≤t∗. 16 On the other hand, note that (19) implies that S(t+T, t) is Lipschitz with constant ec5T, for all T > 0. 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