Exponential attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity

Abstract: In this paper, the dynamical behavior of the nonclassical diffusion equation is investigated. First, using the asymptotic regularity of the solution, we prove that the semigroup {S (t)}t≥0 corresponding to this equation satisfies the global exponentially κ−dissipative. And then we estimate the upper bound of fractal dimension for the global attractors A for this equation and A ⊂ H1 0(Ω) ∩ H2(Ω). Finally, we confirm the existence of exponential attractors M by validated differentiability of the semigroup {S (t)}t≥0. It is worth mentioning that the nonlinearity f satisfies the polynomial growth of arbitrary order.


Introduction
In this paper, we consider the following nonclassical diffusion equation on a bounded domain Ω ⊂ R n with smooth boundary ∂Ω: (1.1) The problem is supplemented with initial data u(x, 0) = u 0 (x), x ∈ Ω, (1.2) and the boundary condition u(x, t)| ∂Ω = 0, for all t ∈ R + , where ν is a positive constant and g = g(x) ∈ L 2 (Ω). For nonlinearity, we always assume that f (s) ∈ C 1 (R, R), f (0) = 0, (1.4) and it satisfies the following conditions: For any s ∈ R, where α, γ, β and δ are positive constants given, and there is a positive constant l such that f (s) ≥ −l. (1.6) This equation appears as an extension of the usual diffusion equation in fluid mechanics, solid mechanics and heat conduction theory (see e.g., [1][2][3]). The Eq (1.1) with a one-time derivative appearing in the highest order term is called pseudo-parabolic or Sobolev-Galpern equation [4]. The existence of global attractors or uniform attractors for this equation has been considered in many monographs and lectures (see e.g., [5][6][7][8][9][10][11][12][13] and the references therein). As nonlinearity satisfies arbitrary polynomial growth condition, the asymptotic behavior of the solution for the nonclassical diffusion equation, especially the existence of exponential attractors, has received considerably less attention in the literature. In some cases similar to Eq (1.1) for some recent results on this equation, the reader can refer to [14][15][16][17].
In recent years, the existence of exponential attractors for different types of evolution models has been studied by many works of literature (see, e.g., [18][19][20][21] and references therein). Generally, exponential attractors can be constructed for dissipative systems which possesses a certain kind of smoothing property. Actually, not only does the smoothing property provide us with an exponential attractive compact set M (i.e. the exponential attractivity of the semigroup), but also it ensures the finite dimensionality of this set. In order to obtain the smoothing property of dissipative systems, we need split the solution of our problem into two parts, one part exponentially decay, and the other part is in some suitable phase spaces with higher regularity (for example, the domain of a suitable fractional power of the operator ∆). For our problem, we note that the two terms which are ∆∂ t u and the nonlinearity f make problem (1.1) differ from usual reaction-diffusion equations or wave-type equations. For the Eq (1.1), if initial data belongs to H 1 0 (Ω), then its solution is always in H 1 0 (Ω), and has no higher regularity, that is similar as hyperbolic equations. Furthermore, when "n > 4" the imbedding D(A) → L ∞ (Ω) is not true, so it's very difficult to obtain the squeezing property for the semigroup {S (t)} t≥0 associated with this equations. These characters cause some difficulties in studying the existence and the regularity of exponential attractors for equation (1.1) when the nonlinearity f satisfies the polynomial growth of arbitrary order and f ∈ C 1 . For the limit of our knowledge, the existence and the regularity of exponential attractors of equation (1.1) is still not confirmed when the nonlinearity f satisfies (1.4)-(1.6).
The main purpose in this paper is to consider the existence of exponential attractors for Eq (1.1). In particular, by verifying the asymptotic regularity of global weak solutions of problem (1.1), we also obtain the regularity of the exponential attractor M , i.e. M ⊂ D(A) with u 0 ∈ H 1 0 (Ω). First, to obtain the finite fractal dimension of global attractors in H 1 0 (Ω), we verify the asymptotic regularity of the semigroup of solutions corresponding to problem (1.1) by using a new decomposition method (or technique) as in [22]. It is worth noticing that authors only proved the the existence of global attractors (autonomous) or uniform attractors (non-autonomous) under a polynomial growth nonlinearity in [22][23][24]. Second, to obtain an exponential attractor in H 1 0 (Ω), we prove that the semigroup is Fréchet differentiable on H 1 0 (Ω) . Obviously, the result obtained in this paper essentially improve and complement earlier ones in [25,26] with critical nonlinearity. This paper is organized as follows. In section 2, we recall some basic concepts as to exponential attractors and useful results that will be used later. In section 3, by using the ideas in [22], we first verify the asymptotic regularity of the semigroup for problem (1.1). Then we obtain the existence and regularity of global attractors of problem (1.1). Finally, the existence and regularity of exponential attractor is proved by using the ideas in [27].

Preliminaries
For conveniences, hereafter let |u| be the modular (or absolute value) of u, |Ω| be the measure of the bounded domain Ω ⊂ R n , | · | p be the norm of L p (Ω)(1 ≤ p ≤ ∞) and (·, ·) be the inner product of L 2 (Ω). C denotes any positive constant which may be different from line to line even in the same line. For the family of Hilbert spaces D(A s+1 2 ), s ≥ 0, their inner products and norms are respectively, Let X be a complete metric space. A one-parameter family of (nonlinear) mappings S (t) : X → X(t ≥ 0) is called semigroup provided that: (1)S (0) = I; (2)S (t + s) = S (t)S (s) for all t, s ≥ 0. Furthermore, we say that semigroup {S (t)} t≥0 is a C 0 semigroup or continuous semigroup if S (t)x 0 is continuous in x 0 ∈ X and t ∈ R.
The pair (S (t), X) is usually referred to as a dynamical system. A set A ⊂ X is called the global attractor for A set B 0 is called a bounded absorbing set for (S (t), X) if for any bounded set B ⊂ X, there exists Lemma 2.1. [19,20,28] A continuous semigroup {S (t)} t≥0 has a global attractor A if and only if S (t) has a bounded absorbing set B and for an arbitrary sequence of points x n ∈ B(n = 1, 2, · · · .), the sequence {S (t n )x n } ∞ n=1 has a convergence subsequence in B. In fact, we know that Now, we briefly review the basic concept of the Kuratowski measure of noncompactness and restate its basic property, which will be used to characterize the existence of exponential attractors for the dynamical system (S (t), X) (the readers refer to [29,30] for more details).
Let X be a Banach space and B be a bounded subset of X. The Kuratowski measure of noncompactness κ(B) of B is defined by κ(B) = inf{δ > 0|B admits a finite cover by sets of diameter ≤ δ}.
are non-empty closed sets in X such that κ(F n ) → 0 as n → ∞ , then F = ∞ n=1 F n is nonempty and compact. In addition, let X be an infinite dimensional Banach space with a decomposition X = X 1 ⊕ X 2 and let P : X → X 1 , Q : X → X 2 be projectors with dim X 1 < ∞. Then (6)κ(B(ε)) = 2ε, where B(ε) is a ball of radius ε; (7)κ(B) < ε, for any bounded subset B ⊂ X for which the diameter of QB is less than ε.
Definition 2.2. [19,28] A semigroup S (t) is called ω-limit compact if for every bounded subset B of X and for any ε > 0, there exists a t 0 > 0 such that where κ is the Kuratowski measure of noncompactness.
Let X be a Banach space with the following decomposition and denote projections by P : X → X 1 and (I − P) : X → X 2 . In addition, let {S (t)} t≥0 be a continuous semigroup on X. Using the idea in [29], the concepts "Condition (C * )" is introduced in [27].
Condition (C * ): For any bounded set B ⊂ X, there exist positive constants t 0 , k and l, such that for any ε > 0, there is a finite dimensional subspace Then there exists a exponential attractor M with the finite fractal dimension.

Priori estimates
The following general existence and uniqueness of solutions for the nonclassical diffusion equations can be obtained by the Galerkin approximation methods, here we only formulate the results: Lemma 3.1. [22,32] Assume that g ∈ L 2 (Ω), and f satisfies (1.4)-(1.6). Then for any initial data u 0 ∈ H 1 0 (Ω) and any T > 0, there exists a unique solution u for the problem (1.1)-(1.2) which satisfies Moreover, we have the following Lipschitz continuity: For any u i 0 (u i 0 ∈ H 1 0 (Ω), denote by u i (i = 1, 2) the corresponding solutions of Eq (1.1), then for all t ≥ T where Q(·) is a monotonically increasing function.
Combining with (3.1), we know that S (t) maps the bounded set of H 1 0 (Ω) into a bounded set for all t 0, that is (Ω) and B be any bounded subset H 1 0 (Ω), then there exists a positive constant χ which depends only on |g| 2 and ρ 0 , such that for any u 0 ∈ B, the following estimate |u(t)| p p ≤ χ, holds for any t ≥ T 0 (from Lemma 3.2).
For brevity, in the sequel, let B 0 be the bounded absorbing set obtained in Lemma 3.2 and let (Ω) and B be any bounded subset of H 1 0 (Ω), for any u 0 ∈ B, then there exist positive constants C which depends on B 0 , such that Proof. Multiplying (1.1) by u, and then integrating in Ω, it now follows that Using the assumptions (1.4)-(1.6), we have We can infer from (3.8) that for any τ > 0, there exists τ 0 ∈ (0, τ] such that Multiplying (1.1) by u t (t) and integrating in Ω, we have Similarly, for any τ > 0, there exists τ 1 ∈ (0, τ] such that |u t (τ 1 )| 2 2 + u t (τ 1 ) 2 0 ≤ M 3 . (3.14) In order to obtain the estimate about u t , differentiate the first equation of (1.1) with respect to t and let z = ∂ t u, then z satisfies the following equality Multiplying (3.15) by z(t), and integrating in Ω, we have Taking t ≥ τ 1 and integrating (3.16) over [τ 1 , t]. Thus, we obtain Let C = M 3 l max{1,ν} min{1,ν} , then the proof is completed.

Exponential attractors
In the following, we will prove the asymptotic regularity of solutions for the Eq (1.1) with initialboundary conditions (1.2)-(1.3) in H 1 0 (Ω) by using a new decomposition method (or technique). In order to obtain the regularity estimates later, we decompose the solution S (t)u 0 = u(t) into the sum: where S 1 (t)u 0 = v(t) and S 2 (t)u 0 = ω(t) solve the following equations respectively, and where the constant µ > 2l max{β, 1} given, l is from (1.6).
Remark 3.6. It is easy to verify the existence and uniqueness of the decomposition (3.17) corresponding to (3.18) and (3.19).
In fact, we can rewrite (3.19) as the following where u is the unique solution of Eq (1.1) with (1.2), so g + µu ∈ L 2 loc (R + , L 2 (Ω)) is known. The existence and uniqueness of solutions ω corresponding to Eq (3.20) can be obtained by the Galerkin approximation methods(see e.g., [19]). By the superposition principle of solutions of partial differential equations, the existence and uniqueness of solutions v for Eq (3.18) can be proved.
We will establish some priori estimates about the solutions of Eqs (3.18) and (3.19), which are the basis of our analysis. The proof is similar to [24]. We also note that this proof was mentioned in [22].  6) and B be any bounded set of H 1 0 (Ω). Assume that S 1 (t)u 0 = v(t) is the solutions of (3.18) with initial data v(0) = u 0 ∈ B. Then, there exists a positive constant d 0 which only depend on l, µ and ν such that for every t ≥ 0 holds, where k 0 = k 0 ( u 0 ) > 0 is a monotonically increasing continuous function about u 0 .
Proof. Multiplying (3.18) by v(t), and integrating in Ω, we obtain By assumptions (1.4)-(1.6), we have It follows that 1 2 then we have d dt |v| 2 2 + ν|∇v| 2 2 + d 0 |v| 2 2 + ν|∇v| 2 2 ≤ 0. By the Gronwall Lemma, for all t ≥ 0, we have the following estimation Next, we will consider the asymptotic regularity of the solution u(t) for (1.1), that is to verify the regularity of the solution ω(t) for Eq (3.19). Concerning the solution ω to Eq (3.19), we have the following result, which shows asymptotic regularity of the solution u to Eq (1.1) with the initialboundary conditions (1.2)-(1.3).
Therefore, for all t ≥ T 0 , we have where M B from Corollary 3.3. Furthermore, by (3.26), we obtain also For any t ≥ T 0 , it follows that t+1 t |ω(s)| p p ds ≤ and then for all t ≥ T * 0 , we have that Multiplying (3.19) by −∆ω(t) and integrating in Ω, we obtain ∆ω).
Remark 3.9. By the proof of the Lemma 3.7 and the Lemma 3.8, we find that the existence and regularity of global attractor A also can be proved under g ∈ H −1 (Ω).
If we take an element of L 2 (Ω) and project it onto the space spanned by the first m eigenfunctions {w 1 , w 2 , · · · , w m }, we get We also define the projection orthogonal of P m , Q m = I − P m , Let u 1 = P m u, u 2 = Q m u = (I − P m )u, then u = u 1 + u 2 and it follows that |(u, w j )| 2 ≤ |u| 2 2 , for any u ∈ L 2 (Ω); (3.34) λ j |(u, w j )| 2 ≥ λ m |u 2 | 2 2 , for any u ∈ H 1 0 (Ω). provided that t ≥ T 2 . This proof is completed. It follows from the conclusions in Lemma 2.7 and T heorem 3.10 that the semigroup {S (t)} t≥0 associated with the Eq (1.1) satisfies the globally exponential κ-dissipative, then the semigroup has a compact and positive invariant set M , which attracts any bounded subset B ⊂ H 1 0 (Ω) exponentially. Next, we are aiming to prove that the fractal dimension of the set M is finite, to this end, the following result is necessary. Proof. By Lemma 3.8, we just need to verify that the fractal dimension of global attractor A is finite. It's obvious A ⊂ D(A) for all t ≥ 0. Take T > 0 fixing and let S n = S (nT ), obviously S n is a discrete dynamical system. The measure of non-compactness is exponentially decaying for S n . Let θ = e −lT and r = kθ (l, k from Lemma 3.10). Since A is compact, for r there exist x 1 , x 2 , · · · , x N such that Then there exist Nq open balls with radius kθ 2 in H 1 0 (Ω) covering A . For any n ∈ N, after iterations, we obtain that there exist at most Nq n−1 balls with radius kθ n in H 1 0 (Ω) covering A . So for all ε > 0, let n ≥ [ ln k−ln ε lT ] + 1, then kθ n = ke −nlT < ε. We get This proof is completed.
Lemma 3.12. For any t > 0, the semigroup {S (t)} t≥0 is Fréchet differentiable on H 1 0 (Ω). Proof. Let S (t * ) u 0 + hv 0 = v(t * ) and S (t * ) u 0 = u(t * ) be the solutions at the time t * for the following equations respectively, And then, setting , which clearly satisfies the following equation h and 0 < θ < 1. Multiplying Eq (3.42) by ω h and integrating over Ω, we have Then ω h (t * ) ∈ H 1 0 (Ω) and where C is a constant independent of h. On the other hand, we denote W = W(x, t) which satisfies the following equation Then where C is a constant independent of h.
The homogenization of the above Eq (3.46) gives It is obvious U ≡ 0 for the homogenization Eq (3.47).

Conclusions
This paper mainly investigate the long-time behavior for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity, including the following three results: (i) the existence and regularity of global attractors is obtained, it is worth noting that a new operator decomposition method is proposed; (ii) the global attractors have finite fractal dimension by combining with asymptotic regularity of solutions; (iii) we confirm the existence of exponential attractors by verifying Fréchet differentiability of semigroup. The above conclusions are more general, and essentially improve existing some results, it should be pointed out that these methods in this paper can also be used for other evolution equations.