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In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.

In this paper, we are concerned with the existence of global attractor and Hausdorff and Fractal dimensions estimation for the following nonlinear Higher-order Kirchhoff-type equations:

where

Recently, Marina Ghisi and Massimo Gobbino [

where

Yang Zhijian, Ding Pengyan and Lei Li [

where

and the nonlinearity

that is,

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

and possesses a weak global attractor.

Yang Zhijian, Ding Pengyan and Liu Zhiming [

where

Li Fucai [

where

Their main results are the two theorems:

Theorem 1. Suppose that

Theorem 2. Suppose that

Li Yan [

where

where

At last, Li Yan studied the asymptotic behavior of solutions for problem (1.14) - (1.16).

For the most of the scholars represented by Yang Zhijian have studied all kinds of low order Kirchhoff equations and only a small number of scholars have studied the blow-up and asymptotic behavior of solutions for higher-order Kirchhoff equation. So, in this context, we study the high-order Kirchhoff equation is very meaningful. In order to study the high-order nonlinear Kirchhoff equation with the damping term, we borrow some of Li Yan’s [

that the equation has a unique smooth solution

and obtain the solution semigroup

For more related results we refer the reader to [

For convenience, we denote the norm and scalar product in

According to [

(H_{1}) Setting

(H_{2}) If

where

(H_{3}) There exist constant

(H_{4}) There exist constant

where

For every_{1})-(H_{3}) and apply Poincaré inequality, there exist constants

where

Lemma 1. Assume (H_{1})-(H_{3}) hold, and

where

is the first eigenvalue of

Proof. We take the scalar product in

After a computation in (2.10), we have

Collecting with (2.11) - (2.14), we obtain from (2.10) that

Since

equality Young’s inequality and Poincaré inequality, we deal with the terms in (2.15) one by one as follow:

By (2.7), we can obtain

where

Because of

By (2.16) - (2.19), it follows from that

By Young’s inequality and

By (2.22), we get

where

By (2.21) and substituting (2.23) into (2.20), we receive

Since

By (2.6) and (2.21), we have

where

Combining with (2.25) and (2.26), formula (2.24) into

We set

where

From conclusion (2.26), we know

where

By generalized Young’s inequality, we have

Then, we get

By (2.26) and (2.30), we have

Combining with (2.29) and (2.31),we obtain

Then,

So, there exist

Lemma 2. In addition to the assumptions of Lemma 1, (H_{1}) - (H_{4}) hold. If (H_{5}):

where

and

Proof. Taking L^{2}-inner product by

After a computation in (2.37) one by one, as follow

By Young’s inequality, we get

Next to estimate _{4}):

By

Collecting with (2.43), from (2.41) we have

By

Integrating (2.38) - (2.40), (2.44) - (2.45), from (2.37) entails

By Poincaré inequality, such that

First, we take proper

to

By Gronwall’s inequality, we get

On account of Lemma 1, we know

Substituting (2.50) into (2.47), we receive

Taking

where

where

Let

Then

So, there exists

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

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

Assume

By multiplying (3.2) by

Exploiting (3.4) - (3.6), we receive

In (3.7), according to Lemma 1 and Lemma 2, such that

where

By (H_{4}), we obtain

where

From the above, we have

For (3.10), because

where

wall’s inequality for (3.11), we obtain

Hence , we can get

That is

Therefore

So we get the uniqueness of the solution.

Theorem 3.2. [

1)

2) It exists a bounded absorbing set

where

3) When

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

where

is the bounded absorbing set of

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

We rewrite the problems (1.1) - (1.3):

Let

Let

mapping

Lemma 4.1 [

1)

2) If

The proof of lemma 4.1 see ref. [

Theorem 4.1. [

Proof. Firstly, we rewrite the equations (4.1), (4.2) into the first order abstract evolution equations in

Let

where

where

We take

For a given time

standard orthogonal basis of the space

From the above, we have

where

where

Now, suppose that

Then there is a

where

Almost to all t, making

So

Let us assume that

According to (4.19), (4.20), so

Therefore, the Lyapunov exponent of

From what has been discussed above, it exists

According to the reference [

In this paper, we prove that the higher-order nonlinear Kirchhoff equation with linear damping in

The authors express their sincere thanks to the aonymous 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 11561076.

Gao, Y.L., Sun, Y.T. and Lin, G.G. (2016) The Global Attractors and Their Hausdorff and Fractal Di- mensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping. International Jour- nal of Modern Nonlinear Theory and Appli- cation, 5, 185-202. http://dx.doi.org/10.4236/ijmnta.2016.54018