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In this paper, we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation . By a priori estimation, we first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the global attractors of the equation.

In 1883, Kirchhoff [

properties of the rope, and

Masamro [

where

Sine-Gordon equation is a very useful model in physics. In 1962, Josephson [

Based on Kirchhoff and Sine-Gordon model, we study the following initial boundary value problem:

where

The rest of this paper is organized as follows. In Section 2, we first obtain the basic assumption. In Section 3, we obtain a priori estimate. In Section 4, we prove the existence of the global attractors.

For brevity, we define the Sobolev space as follows:

In addition, we define

Nonlinear function

Function

Lemma 3.1. Assuming the nonlinear function

where

Proof. Let

Taking the inner product of the equations (3.1) with v in H, we find that

By using Holder inequality, Young’s inequality and Poincare inequality, we deal with the terms in (3.2) one by as follows

where

Since

and

where

Combined (3.1)-(3.6) type, it follows from that

According to condition (F) (5), this will imply

that is

With (3.10), (3.8) can be written as

Set

where

By using Gronwall inequality, we obtain

Let

So, we have

then

Hence, there exists

Lemma 3.2. Assuming the nonlinear function

where

Proof. The equations (3.1) in the H and

By using Holder inequality, Young’s inequality and Poincare inequality, we get the following results

According to condition (F) (5), (6), we obtain

where

By (3.18)-(3.22), (3.17) can be written

Noticing

Substituting (3.24) into (3.23), we can get the following inequality

Let

then

where

By using Gronwall inequality, we obtain

taking

then

Hence, there exists

Theorem 3.1. Assuming the nonlinear function

Proof. By Lemma 3.1-Lemma 3.2 and Glerkin method, we can easily obtain the existence of solutions of equ-

ation

detail.

Assume

We take the inner product of the above equations (3.31) with

We deal with the terms in (3.32) one by as follows

and

By (3.32)-(3.34), we can get the following inequality

Further, by mid-value theorem and Young’s inequality, we get

Since

might as well set

where

Then, we obtain

Substituting (3.36), (3.37) into (3.35), we can get

Let

By using Gronwall inequality, we obtain

There has

That show that

So as to get

Theorem 4.1. [

1)

2) It exists a bounded absorbing set

here

3) When

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

Theorem 4.2. [

where

(1)

(2)

Proof. Under the conditions of Theorem 3.1, it exists the solution semigroup S(t), here

(1) From Lemma 3.1-Lemma 3.2, we can get that

This shows that

(2) Furthermore, for any

So we get

(3) Since

Hence, the semigroup operator S(t) exists a compact global attractor A. The proving is completed.

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.

RuijinLou,PenghuiLv,GuoguangLin, (2016) Global Attractors for a Class of Generalized Nonlinear Kirchhoff-Sine-Gordon Equation. International Journal of Modern Nonlinear Theory and Application,05,73-81. doi: 10.4236/ijmnta.2016.51008