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In this paper, we study the long time behavior of solution to the initial boundary value problem for a class of Kirchhoff-Boussinesq model flow . We first prove the wellness of the solutions. Then we establish the existence of global attractor.

In this paper, we are concerned with the existence of global attractor for the following nonlinear plate equation referred to as Kirchhoff-Boussinesq model:

where

Recently, Chueshov and Lasiecka [

with clamped boundary condition

with

When

which Yang Zhijian and Jin Baoxia [

with

Zhijian Yang, Na Feng and Ro Fu Ma [

In this model, g satisfies the nonexplosion condition,

where

T. F. Ma and M. L. Pelicer [

with simply supported boundary condition

and initial condition

where

For more related results we refer the reader to [

to make these equations more normal; we try to make a different hypothesis (specified Section 2), by combining the idea of Liang Guo, Zhaoqin Yuan, Guoguang Lin [

For brevity, we use the follow abbreviation:

with

In this section, we present some materials needed in the proof of our results, state a global existence result, and prove our main result. For this reason, we assume that

(H_{1})

where

where

(H_{2})

Now, we can do priori estimates for Equation (1.1).

Lemma 1. Assume (H_{1}), (H_{2}) hold, and

where

Remark 1. (2.1) and (2.1) imply that there exist positive constants

Proof of Lemma 1.

Proof. Let

Taking H-inner product by v in (2.7), we have

Since

Poincare inequality, we deal with the terms in (2.8) one by one as follow,

and

By (2.9)-(2.13), it follows from that

By (2.6), we can obtain

Substituting (2.15) into (2.14), we receive

By using Holder inequality, Young’s inequality, and (H_{2}), we obtain

Then, we have

Because of

Substituting (2.20) into (2.19) gets

Taking

where

From (H_{1}):

Then

So, there exists

■

Lemma 2. In addition to the assumptions of Lemma 1, if (H_{3}):

where

Proof. Taking H-inner product by

Using Holder inequality, Young’s inequality and Poincare inequality, we deal with the terms in (2.29) one by one as follow,

and

Substituting (2.30)-(2.33) into (2.29), we can obtain that

By using Holder inequality, Young’s inequality, and (H_{1}), (H_{3}), we obtain

By using Gagliardo-Nirenberg inequality, and according the Lemma 1, we can get

By using the same inequality, we can obtain

By using Gagliardo-Nirenberg inequality, and according the Lemma 1, we can get

where

Substituting (2.35), (2.37), (2.40) into (2.34), we receive

Because of

Taking

where

Let

Then

So, there exists

■

Theorem 3.1. Assume that

where

where

Then the problem (1.1)-(1.3) exists a unique smooth solution

Remark 2. We denote the solution in Theorem 3.1 by

Proof of Theorem 3.1.

Proof. By the Galerkin method and Lemma 1, we can easily obtain the existence of Solutions. Next, we prove the uniqueness of Solutions in detail. Assume

Taking H-inner product by

By (H_{1}), (H_{2})

where

By using Gagliardo-Nirenberg inequality, and according the Lemma 1,we can get

Substituting (3.3), (3.5) into (3.2)

Taking

Then

By using Gronwall inequality, we obtain

So, we can get

That shows that

That is

Therefore

We get the uniqueness of the solution. So the proof of the Theorem 3.1. has been completed. ■

Theorem 3.2. [

1)

2) It exists a bounded absorbing set

here

3) When

Therefore, the semigroup operators S(t) exist a compact global attractor A.

Theorem 3.3 Under the assume of Theorem 3.1, equations have global attractor

where

1)

2)

Proof. Under the conditions of Theorem 3.1, it exists the solution semigroup

(1) From Lemma 1-Lemma 2, we can ge that

This shows that

(2) Furthermore, for any

So we get

(3) Since

The authors express their sincere thanks to the anonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. These contributions greatly improved the paper.

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11161057.

PenghuiLv,RuijinLou,GuoguangLin, (2016) Global Attractor for a Class of Nonlinear Generalized Kirchhoff-Boussinesq Model. International Journal of Modern Nonlinear Theory and Application,05,82-92. doi: 10.4236/ijmnta.2016.51009