^{1}

^{1}

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Under the assumption that g(t) is translation bounded in , and using the method developed in [3], we prove the existence of pullback exponential attractors in for nonlinear reaction diffusion equation with polynomial growth nonlinearity( is arbitrary).

Attractor’s theory is very important to describe the long time behavior of dissipative dynamical systems generated by evolution equations, and there are several kinds of attractors. In this article, we will study the existence of pullback exponential attractors (see [

where

for all

The Equation of (1.1) has been widely studied. For the autonomous case, i.e.,

In order to obtain the existence of pullback exponential attractors of (1.1), we will need the following theorem.

Theorem 1.1. ([

(1) There exists an uniformly bounded absorbing set

(2) There exist

for all

In this section, we will derive some priori estimates for the solutions of (1.1) that will be used to construct pullback exponential attractors for the problem (1.1).

For convenience, hereafter let

For the initial value problem (1.1), we know from [

Thanks to the existence theorem, the initial value problem is equivalent to a process

In addition, we assume that the function

By (2.1), for

Lemma 2.1. ([

and

Lemma 2.2. Assume that

Obviously, for any bounded

Proof. Let

Taking inner product of (1.1) with

Multiply (1.1) by

since

Combining (2.7), we get

Thanks to Poincaré inequality

Let

which imply

integrating, we get

using (2.3) and (2.4), we get the inequality (2.5).

Lemma 2.3. Assume that

Here

By the assumption (2.1) and for

Proof. Multiply (1.1) with

By (1.2) and Young’s inequality, we have

By (2.13), we get

integrating and using (2.4), we get

Multiply (1.1) with

By (2.1), we get

Using Young’s inequality

By the above inequality, we have

integrating and using (2.12) and (2.14), we get (2.11) holds.

Lemma 2.1, lemma 2.2 and lemma 2.3 show that the process generated by the equation (1.1) have an uniformly pullback bounded absorbing set in

Theorem 2.4. Assume that

In fact, using the same proof as in Lemma 2.3, we can get the following result.

Lemma 2.5. Assume that

an uniformly pullback bounded absorbing set

In this section, we will use Theorem 1.1 to prove that the process generated by Equation (1.1) exists a pullback exponential attractor.

First we assume that the function

Obviously,

We set

Theorem 2.4. Assume that

Next, we will verify that the process generated by (1.1) satisfy all the conditions of Theorem 1.1.

Proof. By Theorem 2.4, there exists

We set

Taking inner product of (3.2) with

Taking into account (1.2) and Holder inequality, it is immediate to see that

and

By Lemma 2.5, we get

Using (3.3), we obtain

Let

Taking into (3.4) account, we obtain

Using the Poincaré inequality

Let

Since

By Gronwall’s lemma, we get

By (3.1), we obtain that there exists

Let

Since

By Theorem 2.4 and (3.9)-(3.11), we know that the process

This work was supported by the National Nature Science Foundation of China (11261027) and Longyuan youth innovative talents support programs of 2014, and the innovation Funds of principal (LZCU-XZ2014-05).

Yongjun Li,Yanhong Zhang,Xiaona Wei, (2015) Pullback Exponential Attractors for Nonautonomous Reaction Diffusion Equations in H_{0}^{1}. Journal of Applied Mathematics and Physics,03,730-736. doi: 10.4236/jamp.2015.37087