The Global Attractors for the Higher-Order Kirchhoff-Type Equation with Nonlinear Strongly Damped Term

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 (H1) - (H4). 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.

1. Introduction

We consider the following Higher-order Kirchhoff-type equation: (1.1) (1.2) (1.3)

where is an integer constant, and is a bounded domain of , with a smooth dirichlet boundary and initial value. Moreover, is the unit outward normal on . and are scalar functions specified later, f is a given function.

This kind of wave models goes back to G. Kirchhoff  and has been studied by many authors under different types of hypotheses. There have been many researchers on the global attractors existence of Kirchhoff equation, we can refer      . What’s more, the global attractors for the Higher-order Kirchhoff-type equation are investigated and we refer to    .

Zhijian Yang and Pengyan Ding  studied the longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on : (1.4)

They establish the well-posedness, the existence of the global and exponential attractors in natural energy space in critical nonlinearity case. On this basis, they also investigated the global well-posedness and the longtime dynamics of the Kirchhoff equation with fractional damping and supertical nonlinearity  :

(1.5)

The main results are focused on the relationships among the growth exponent p of the nonlinearity, the global well-posedness and the longtime dynamics of the equations. They show that i) even if p is up to the supercritical range, that is,

, the well-posedness and the longtime behavior of the solutions of

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

, the corresponding subclass G of the limit solutions exists

and possesses a weak global attractors.

Varga Kalantarov and Sergey Zelik  present a new method of investigating the so-called quasi-linear strongly damped wave equations:

(1.6)

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 is less than 6 and f may have arbitrary polynomial growth rate. Moreover, the existence of a finite-dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established. In a particular case which corresponds to the so-called semi-linear strongly damped wave equation, their result allows to remove the long-standing growth restriction . The Cauchy problem and the boundary value problem for equation under the different assumptions on the nonlinearities and f have been studied in many papers, but the author uses a new method to this equation.

Xiuli Lin and Fushan Li  consider the initial-boundary value problem for nonlinear Kirchhoff-type equation:

(1.7)

where and are constants, is a -function such that for all. Under suitable conditions on the initial data, they show the existence and uniqueness of global solution by means of the Galerkin method and the uniform decay rate of the energy by an integral inequality. Here, satisfying and. In this paper, for strong nonlinear damping and, we make some similar assumptions. These assumptions will be presented in the following statements.

In 2004, Fucai Li  dealed with the higher-order Kirchhoff-type equation with nonlinear dissipation:

(1.8)

In a bounded domain, where is a positive integer, and are positive constants. They obtain that the solution exists global if, while if , then for any initial data with negative initial energy, the solution blows up at finite time in norm.

In 2007, Salim A. Messaoudi and Belkacern Said Houari  improve Li’s result and showed that certain solutions with positive initial energy also blow up in finite time.

Qingyong Gao, Fushan Li, Yanguo Wang  obtained the local existence of the solution to the homogeneous Dirichlet boundary value problem for the higher-order nonlinear Kirchhoff-type equation:

(1.9)

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  studied the longtime dynamics of Kirchhoff wave models with strong nonlinear damping:

(1.10)

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 is a definite constant and even . Generally, the equation exist a nonlinear. But in the paper, is a scalar function and. Under of the the proper assume, in

section 2, we prove the existence of the solution by priori estimation and the Galerkin method. Therefore, we show that i) the solution of the problem (1.1) - (1.3) satisfies; further more, ii) the solution of the problem (1.1) - (1.3) satisfies. Then, in section 3, we prove the uniqueness of the solution by using the method that assumption exist two solutions in the same initial value and two solutions are equal. At last, according to define, we obtain to the existence of the global attractor.

2. Preliminaries

For brevity, we denote the simple symbol, represents inner product, and , , , , , , , are constants, are also constants. is the first eigenvalue of the operator.

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

(H1) setting, then

(2.1)

where.

(H2) 

(2.2)

(H3)

(2.3)

(H4)

(2.4)

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

Lemma 1. Assume (H1) hold, and,. Then the solution of the problem (1.1) - (1.3) satisfies, and

(2.5)

where, ,. Thus, there exists and, such that

(2.6)

Proof. Let, then we use v multiply with both sides of Equation (1.1) and obtain

(2.7)

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

(2.8)

(2.9)

(2.10)

Because, by using Holder inequality, Young’s inequality, we obtain

(2.11)

From the above, we have

(2.12)

According to (2.1), we have

(2.13)

where.

Substitution (2.13) into (2.12), we receive

(2.14)

We deal with the items, we have

(2.15)

where we take a proper constant, such that

Then, we get

(2.16)

where

(2.17)

By using Gronwall inequality, we obtain

(2.18)

where

(2.19)

So, we have

(2.20)

and

(2.21)

Thus, there exist and, such that

(2.22)

Remark 1. Assumption (H1) imply

(2.23)

such that (2.20) hold.

Lemma 2. Assume (H2) hold, , and. Then the solution of the problem (1.1) - (1.3) satisfies, and

(2.24)

where, , ,. There exist and, such that

(2.25)

Proof. Let, we use multiply sides of equation (1.1) and obtain

(2.26)

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

(2.27)

(2.28)

(2.29)

Due to, by using Holder inequality, Young’s inequality, we obtain

(2.30)

From the above, we obtain

(2.31)

According to (2.2), we have

(2.32)

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

(2.33)

Noticing, this will imply

(2.34)

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

(2.35)

Hence, we take a proper constant, such that, we get

(2.36)

where

(2.37)

By using Gronwall inequality, we end up with

(2.38)

where

(2.39)

Taking, we have

(2.40)

and

(2.41)

Thus, there exist and, such that

(2.42)

3. Global Attractor

3.1. The Existence and Uniqueness of Solution

Theorem 3.1. Assume (H1) - (H4) hold, and, ,. So equality (1.1) exists a unique smooth solution.

Remark 2. We denote the solution in Theorem 3.1 by. Then composes a continuous semigroup in.

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 are two solutions of the problems (1.1) - (1.3), we denote, then, and the two equations subtract and obtain

(3.1)

By using to inner product of the equation (3.1), and we have

(3.2)

(3.3)

(3.4)

Next, we process each item in turn

(3.5)

Analogous to, we deal with

(3.6)

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

(3.7)

Similarly,

(3.8)

Therefore, by the above inequality

(3.9)

when, we get

(3.10)

In view of (H4), there exist constant, and let, such that

(3.11)

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

(3.12)

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

(3.13)

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

(3.14)

Supposing there is a constant K large enough, we have

(3.15)

where,.

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

Due to (3.14), we have

(3.16)

where

(3.17)

Therefore,

(3.18)

where

(3.19)

So, we can get

(3.20)

According to (3.12), we get

(3.21)

That shows that

(3.22)

That is

(3.23)

Therefore,

(3.24)

So we prove the uniqueness of the solution.

3.2. Global Attractor

Theorem 3.2.  Let E be a Banach space, and are the semigroup operator on E., , , here I is a unit operator. Set satisfy the follow conditions:

1) is uniformly bounded, namely, it exists a constant, so that

(3.25)

2) It exists a bounded absorbing set, namely, , it exists a constant, so that

(3.26)

where and are bounded sets.

3) When, is a completely continuous operator A.

Therefore, the semigroup operators exists a compact global attractor A.

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

(3.27)

where

(3.28)

is the bounded absorbing set of and satisfies.

1);

2), here and it is a bounded set,

(3.29)

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

1) From Lemma 1 to Lemma 2, we can get that is a bounded set that includes in the ball,

(3.30)

This shows that is uniformly bounded in.

2) Furthermore, for any, when, we have

(3.31)

So we get is the bounded absorbing set.

3) Since is compact embedded, which means that the bounded set in is the compact set in, so the semigroup operator S(t) exist a compact global attractor A.

The prove is completed.

4. Conclusion

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 of the problem (1.1) - (1.3) satisfies ; furthermore, ii) the solution of the problem (1.1) - (1.3) satisfies. Then, we prove the uniqueness of the solution. At last, according to define and theorem, we obtain to the existence of the global attractor.

Acknowledgements

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.

Fund

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

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

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

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