Exponential Attractors of the Nonclassical Diffusion Equations with Lower Regular Forcing Term

In this article, we prove the existence of exponential attractors of the nonclassical diffusion equation with critical nonlinearity and lower regular forcing term. As an additional product, we show that the fractal dimension of the global attractors of this problem is finite.


Introduction
We consider the asymptotic behavior of solutions to be the following nonclassical diffusion equation: where ( ) is a bounded domain with smooth boundary ∂Ω , and the external forcing term where 0 C is a positive constant and 1 λ is the first eigenvalue of −∆ on ( ) H Ω .The number 2 1 2 N N + − − is called the critical exponent; since the nonlinearity f is not compact in this case, this is one of the essential difficulties in studying the asymptotic behavior.This equation appears as a nonclassical diffusion equation in fluid mechanics, solid mechanics and heat conduction theory, see for instance [1]- [3] and the references therein.
Since Equation (1.1) contains the term t u −∆ , it is different from the usual reaction diffusion equation essentially.For example, the reaction diffusion equations has some smoothing effect, that is, although the initial data only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity.However, for Equation (1.1), if the initial data 0 u belongs to ( ) H Ω , the solution ( ) H Ω and has no higher regularity because of t u −∆ , which is similar to the hyperbolic equation.Consequently, its dynamics would be more complex and interesting.
The long-time behavior of the solutions of (1.1) has been considered by many researchers; see, e.g.[4]- [9], and the references therein.For instance, for the case ( ) ( ) Ω , the existence of a global attractor of (1.1) in ( ) H Ω was obtained in [4] under the assumptions that f satisfies (1.2) and (1.3) corresponding to 3 N = and the additional condition , which essentially requires that the nonlinearity is subcritical.In [7] the authors investigated the existence of the global attractors for ( ) ( ) ∈ Ω , and proved the asymptotic regularity and existence of exponential attractors for , ; only under the conditions (1.2)- (1.3).Recently, the authors in [9] showed the asymptotic regularity of solutions of Equation (1.1) in ( ) and for ( ) ( ) u H ∈ Ω only under the assumptions (1.2)-(1.3).For the limit of our knowledge, the existence of exponential attractors of Equation (1.1) has not been achieved by predecessors for ( ) ( ) On the other hand, we note that in [10] the authors scrutinized the asymptotic regularity of the solutions for a semilinear second order evolution equation when ( ) ( ) ∈ Ω , and based on this regularity, they constructed a family of finite dimensional exponential attractors.However, they require the following additional technical assumptions besides (1.2) and (1.3): , as 3, 4,5; , as 6, for all , In this article, motivated by the work in [10]- [12], based on the asymptotic regularity in [9], we construct a finite dimensional exponential attractor of (1.1)only under the conditions (1.2) and (1.3).
Our main result is Theorem 1.1 Assume ( ) ≥ associated with problem (1.1) has an exponential attractor  in ( ) H Ω , we know that ⊂   , then Theorem 1.1 implies that fractal dimension of the global attractor  is finite.

Notations and Preliminaries
In this section, for convenience, we introduce some notations about the functions space which will be used later throughout this article., , and .

2
L Ω inner product and norm, respectively.
Moreover,the solution continuously depends on the initial data in  .
In the remainder of this section, we denote by ≥ the semigroup associated with the solutions of (1.1)-(1.3).

Lemma 3.2 ([7]
) Under conditions of above Lemma, There is a positive constant ρ such that for any bounded subset B ⊂  , there exists ( ) , for all and .
From this Lemma, we know that the semigroup of operators ( ) ( ) , u t v t be two solutions of (1.1) with ( ) ( ) , respectively, it follows that 0 0 e , for all 0.
Taking the scalar product of (3.3) with w , we find, ( ) From the condition (1.2), by using the Hölder inequality, and noting the embedding And then, by means of (3.1), we obtain So, combining with Equation (3.4), (3.5), we get ( ) ( ) then using the Gronwall lemma to above inequality, we can conclude our lemma immediately.Lemma 3.4 ([9]) Let ( ) Then, for any , there exists a subset σ  , a positive constant η and a monotone increasing function ( ) ( ) where σ  and ( ) for some positive constant σ Λ ; And ( ) is the unique solution of the following elliptic equation where the constant ( ) Ω such that 1 < 1 where the constant  where ( ) x φ is a fixed solution of (3.8), and ( ) , w x t satisfies the following equation : At the same time, noticing the embedding H H σ  , and from Lemma 3.5 we yield ( ) ( ) Taking the inner product of (3.12) with t w , we get , 1 1 Thus, combining with (3.14), there holds  Firstly, for each fixed 0, min , we define ( ) where σ  is the set obtained in Lemma 3.4.Then, from Lemma 3.5 we know that Secondly, let us establish some properties of this set.
•  is positive invariant.In fact, from the continuity of ) (t S , we have • There holds This is a direct consequence of Lemma 3.6.Therefore such a set  is a promising candidate for our purpose.Finally, we need the following two lemmas.Lemma 3.7 For every > 0 T , the mapping ( ) ( )

S t u S t u S t u S t u S t u S t u
The first term of the above inequality is handled by estimate (3.2).Concerning the second one, Therefore, we will have the following lemma: Lemma 3. 8 The following two estimates hold: 2 2 0 0 0 0 , for all 0 , 0 , where the constant K depends only on t * and ( ) where ( ) v t and ( ) w t solve the following equations respectively: and consider the family of Hilbert space ( )

H
Ω when f satisfies (1.2)-(1.3).Lemma 3.5 ([9]) Under the assumption of Lemma 3.4, for any bounded subset 1 B σ ⊂  , if the initial data the solution ( ) u t of (1.1) has the following estimates similar to (3.7) in Lemma 3.4, more precisely, we have

1 BM
depends only on σ and the σ

9
* , we can get the estimate (3.29).Proof of Theorem 1.1 Applying the abstract results devised in[10]-[12], from Lemma 3.7 and Lemma 3.8, we can prove the existence of an exponential attractor  for As a direct consequence of Theorem 1.1 and the a priori estimates given in [[9], Lemma 3.5] and Lemma 3.8, we decompose  as solution of (3.8).