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 is a bounded domain with smooth boundary, and the external forcing term, non-linear function with and satisfies the following conditions:

and

where is a positive constant and is the first eigenvalue of on. The number is called the critical exponent; since the nonlinearity 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, 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 belongs to, the solution with is always in and has no higher regularity because of, 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, the existence of a global attractor of (1.1) in was obtained in [4] under the assumptions that satisfies (1.2) and (1.3) corresponding to

and the additional condition with, which essentially requires that the nonlinearity is subcritical. In [7] the authors investigated the existence of the global attractors for, 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 for any and for

, 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, 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):

and

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 and satisfies (1.2)-(1.3),. Then the semigroup associated with problem (1.1) has an exponential attractor in.

Remark 1.1 If is a global attractor of (1.1) in, we know that, then Theorem 1.1 implies that fractal dimension of the global attractor is finite.

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

• with domain, and consider the family of Hilbert space with the standard inner products and norms, respectively,

Especially, means the inner product and norm, respectively.

• with the usual norm. Especially, we denote and.

• are continuous increasing functions.

• denote the general positive constants, , which will be different from line to line.

We also need the following the transitivity property of exponential attraction, e.g., see [[12] , Theorem 5.1]:

Lemma 2.1 ([13] ) Let be subsets of such that

for some and. Assume also that for all there holds

for some and. Then it follows that

where and.

In this subsection, based on the asymptotic regularity obtained in [9] , we will construct an exponential attractor by the methods and techniques devised in [10] -[12] . We first need the following Lemmas:

Lemma 3.1 ([7] ) Let satisfies (1.2)-(1.3) and. Then for any and any, there is a unique solution of (1.1) such that

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 such that for any bounded subset, there exists such that

From this Lemma, we know that the semigroup of operators generalized by (1.1) possesses a bounded absorbing set in.

Lemma 3.3 Under conditions of, and be two solutions of (1.1) with, respectively, it follows that

Proof Let satisfies the following equation

Taking the scalar product of (3.3) with, we find,

From the condition (1.2), by using the Hölder inequality, and noting the embedding, we have

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 and satisfies (1.2), (1.3),. Then, for any

, there exists a subset, a positive constant and a monotone increasing function

such that for any bounded set,

where and depend on but is independent of; satisfying

for some positive constant; And is the unique solution of the following elliptic equation

where the constant such that. Furthermore, we know that the solution only belongs to when satisfies (1.2)-(1.3).

Lemma 3.5 ([9] ) Under the assumption of Lemma 3.4, for any bounded subset, if the initial data, then the solution of (1.1) has the following estimates similar to (3.7) in Lemma 3.4, more precisely, we have

where the constant depends only on and the -bound of.

Lemma 3.6 There exists such that

Proof For the solution of (1.1), we now decompose as follows

where is a fixed solution of (3.8), and satisfies the following equation :

At the same time, noticing the embedding, and from Lemma 3.5 we yield

Taking the inner product of (3.12) with, we get

By means of (3.1) and (3.13) and together with Hlder, Young inequalities, it follows that

Thus, combining with (3.14), there holds

Integrating the above inequality on and noting, the proof completes.

Next, we will prepared for constructing an exponential attractor of in by applying the abstract results devised in [10] -[12] [14] .

Firstly, for each fixed, we define

where 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 a compact set in, due to Lemma 3.4.

• is positive invariant. In fact, from the continuity of, we have

• There holds

Indeed, it is apparent that

Hence, (3.19) follows from Lemma 2.1.

• There is such that

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, the mapping is Lipschitz continuous on.

Proof For and we have

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

Hence, there exists a constant, such that

On the other hands, for each initial data, we can decompose the solution of (1.1) as

where and solve the following equations respectively:

and

Therefore, we will have the following lemma:

Lemma 3.8 The following two estimates hold:

and

where the constant depends only on and.

Proof Given two solutions of Equation (1.1) origination from, respectively.

Set

where and solve the following equations respectively:

and

It is apparent that and

Taking the product of (3.30) with in, we get

So

Hence, setting

we have

So, we obtain the result (3.28).

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

Since, we have.

So, from (1.2) and using Hlder inequality, we have

where the constant comes from the embedding,.

From Lemma 3.3, we obtain the inequality

and an integration on, 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 in immediately.

Remark 3.9 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, where is bounded in for any

and is the unique solution of (3.8).

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.