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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.

Nonclassical Diffusion Equations; Exponential Attractor; Critical Exponent; Lower Regular Forcing Term

We consider the asymptotic behavior of solutions to be the following nonclassical diffusion equation:

where

and

where

This equation appears as a nonclassical diffusion equation in fluid mechanics, solid mechanics and heat conduction theory, see for instance [

Since Equation (1.1) contains the term

The long-time behavior of the solutions of (1.1) has been considered by many researchers; see, e.g. [

and the additional condition

For the limit of our knowledge, the existence of exponential attractors of Equation (1.1) has not been achieved by predecessors for

and

In this article, motivated by the work in [

Our main result is Theorem 1.1 Assume

Remark 1.1 If

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

•

Especially,

•

•

•

We also need the following the transitivity property of exponential attraction, e.g., see [[

Lemma 2.1 ([

for some

for some

where

In this subsection, based on the asymptotic regularity obtained in [

Lemma 3.1 ([

Moreover,the solution continuously depends on the initial data in

In the remainder of this section, we denote by

Lemma 3.2 ([

From this Lemma, we know that the semigroup of operators

Lemma 3.3 Under conditions of

Proof Let

Taking the scalar product of (3.3) with

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 ([

where

for some positive constant

where the constant

Lemma 3.5 ([

where the constant

Lemma 3.6 There exists

Proof For the solution

where

At the same time, noticing the embedding

Taking the inner product of (3.12) with

By means of (3.1) and (3.13) and together with H

Thus, combining with (3.14), there holds

Integrating the above inequality on

Next, we will prepared for constructing an exponential attractor of

Firstly, for each fixed

where

Secondly, let us establish some properties of this set.

•

•

• There holds

Indeed, it is apparent that

Hence, (3.19) follows from Lemma 2.1.

• There is

This is a direct consequence of Lemma 3.6.

Therefore such a set

Finally, we need the following two lemmas.

Lemma 3.7 For every

Proof For

The first term of the above inequality is handled by estimate (3.2). Concerning the second one,

Hence, there exists a constant

On the other hands, for each initial data

where

and

Therefore, we will have the following lemma:

Lemma 3.8 The following two estimates hold:

and

where the constant

Proof Given two solutions

Set

where

and

It is apparent that

Taking the product of (3.30) with

So

Hence, setting

we have

So, we obtain the result (3.28).

On the other hands, taking the product of (3.31) with

Since

So, from (1.2) and using H

where the constant

From Lemma 3.3, we obtain the inequality

and an integration on

Proof of Theorem 1.1 Applying the abstract results devised in [

Remark 3.9 As a direct consequence of Theorem 1.1 and the a priori estimates given in [[

The authors thank the referee for his/her comments and suggestions, which have improved the original version of this article essentially. This work was partly supported by the NSFC (11061030,11101334) and the NSF of Gansu Province(1107RJZA223), in part by the Fundamental Research Funds for the Gansu Universities.