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We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping:
. For strong nonlinear damping
σ and
?, we make assumptions (H
_{1}) - (H
_{4}). Under of the proper assume, the main
results are existence and uniqueness of the solution in proved by Galerkin method, and deal with the global attractors.

We consider the following Higher-order Kirchhoff-type equation:

where

This kind of wave models goes back to G. Kirchhoff [

Zhijian Yang and Pengyan Ding [

They establish the well-posedness, the existence of the global and exponential attractors in natural energy space

The main results are focused on the relationships among the growth exponent p of the nonlinearity

the equation are the characters of the parabolic equation; ii) when

and possesses a weak global attractors.

Varga Kalantarov and Sergey Zelik [

In bounded 3D domains. This method establishes the existence and uniqueness of energy solutions in the case where the growth exponent of the non-linearity

Xiuli Lin and Fushan Li [

where

In 2004, Fucai Li [

In a bounded domain, where

In 2007, Salim A. Messaoudi and Belkacern Said Houari [

Qingyong Gao, Fushan Li, Yanguo Wang [

where

At present, most Higher-order Kirchhoff-type equations investigate the blow-up of the solution. We study the global attractor of the solution for Higher-order Kirchhoff- type equations.

Igor Chueshov [

He proves the existence and uniqueness of weak solutions, and established a finite- dimensional global attractor in the sense of partially strong topology.

On the basis of Igor Chueshov, we investigate the global attractor of the higher-order Kirchhoff-type Equation (1.1) with strong nonlinear damping. Such problems have

been studied by many authors, but

section 2, we prove the existence of the solution by priori estimation and the Galerkin method. Therefore, we show that i) the solution

For brevity, we denote the simple symbol,

In this section, we present some assumptions needed in the proof of our results. For this reason, we assume that

(H_{1}) setting

where

(H_{2}) [

(H_{3})

(H_{4})

Now, we can do priori estimates for equation (1.1)

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

where

Proof. Let

After a computation (2.7) one by one, as follow

Because

From the above, we have

According to (2.1), we have

where

Substitution (2.13) into (2.12), we receive

We deal with the items, we have

where we take a proper constant

Then, we get

where

By using Gronwall inequality, we obtain

where

So, we have

and

Thus, there exist

Remark 1. Assumption (H_{1}) imply

such that (2.20) hold.

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

where

Proof. Let

After a computation (2.26) one by one, as follow

Due to

From the above, we obtain

According to (2.2), we have

Collecting with (2.32), we obtain from (2.31) that

Noticing

Substituting (2.34) into (2.33), we can get the following inequality

Hence, we take a proper constant

where

By using Gronwall inequality, we end up with

where

Taking

and

Thus, there exist

Theorem 3.1. Assume (H_{1}) - (H_{4}) hold, and

Remark 2. We denote the solution in Theorem 3.1 by

Proof. By the Galerkin method, Lemma 1 and Lemma 2, we can easily obtain the existence of Solutions, the procedure is omitted. Next, we prove the uniqueness of Solutions in detail. Let

By using

Next, we process each item in turn

Analogous to

Combining with (3.5) - (3.6), we obtain from (3.4) that

Similarly,

Therefore, by the above inequality

when

In view of (H_{4}), there exist constant

According to Hölder inequality, Young’s inequality and Poincaré inequality, we obtain

Combining with (3.11) - (3.12), we receive

Next, we prove that there is a constant K large enough, such that

Supposing there is a constant K large enough, we have

where

Hence, there is a constant K large enough, such that (3.14) hold.

Due to (3.14), we have

where

Therefore,

where

So, we can get

According to (3.12), we get

That shows that

That is

Therefore,

So we prove the uniqueness of the solution.

Theorem 3.2. [

1)

2) It exists a bounded absorbing set

where

3) When

Therefore, the semigroup operators

Theorem 3.3. Under the assume of Lemma 1, Lemma 2 and Theorem 3.1, equations have global attractor

where

1)

2)

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

1) From Lemma 1 to Lemma 2, we can get that

This shows that

2) Furthermore, for any

So we get

3) Since

The prove is completed.

The paper’s main results deal with global attractors. At first, we prove the existence and uniqueness of the solution. Then we establish the existence of the global attractors. There- fore, we show that i) the solution

We express our sincere thanks to the anonymous reviewer for his/her careful reading of the paper, we hope that we can get valuable comments and suggestions. These contributions greatly improved the paper, and making the paper better.

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11561076.

Sun, Y.T., Gao, Y.L. and Lin, G.G. (2016) The Global Attractors for the Higher-Order Kirchhoff- Type Equation with Nonlinear Strongly Damped Term. International Journal of Mo- dern Nonlinear Theory and Application, 5, 203-217. http://dx.doi.org/10.4236/ijmnta.2016.54019