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.
Strongly Nonlinear Damped Higher-Order Kirchhoff Equation The Existence and Uniqueness The Global Attractors1. Introduction
We consider the following Higher-order Kirchhoff-type equation:
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 [1] 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 [2] [3] [4] [5] [6] . What’s more, the global attractors for the Higher-order Kirchhoff-type equation are investigated and we refer to [7] [8] [9] .
Zhijian Yang and Pengyan Ding [2] studied the longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on:
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 [3] :
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 [5] present a new method of investigating the so-called quasi-linear strongly damped wave equations:
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 [6] consider the initial-boundary value problem for nonlinear Kirchhoff-type equation:
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 [7] dealed with the higher-order Kirchhoff-type equation with nonlinear dissipation:
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 [8] 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 [9] obtained the local existence of the solution to the homogeneous Dirichlet boundary value problem for the higher-order nonlinear Kirchhoff-type equation:
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 [4] studied the longtime dynamics of Kirchhoff wave models with strong nonlinear damping:
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
where.
(H2) [10]
(H3)
(H4)
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
where, ,. Thus, there exists and, such that
Proof. Let, then we use v multiply with both sides of Equation (1.1) and obtain
After a computation (2.7) one by one, as follow
Because, by using Holder inequality, Young’s inequality, we obtain
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, such that
Then, we get
where
By using Gronwall inequality, we obtain
where
So, we have
and
Thus, there exist and, such that
Remark 1. Assumption (H1) imply
such that (2.20) hold.
Lemma 2. Assume (H2) hold, , and. Then the solution of the problem (1.1) - (1.3) satisfies, and
where, , ,. There exist and, such that
Proof. Let, we use multiply sides of equation (1.1) and obtain
After a computation (2.26) one by one, as follow
Due to, by using Holder inequality, Young’s inequality, we obtain
From the above, we obtain
According to (2.2), we have
Collecting with (2.32), we obtain from (2.31) that
Noticing, this will imply
Substituting (2.34) into (2.33), we can get the following inequality
Hence, we take a proper constant, such that, we get
where
By using Gronwall inequality, we end up with
where
Taking, we have
and
Thus, there exist and, such that
3. Global Attractor3.1. 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
By using to inner product of the equation (3.1), and we have
Next, we process each item in turn
Analogous to, we deal with
Combining with (3.5) - (3.6), we obtain from (3.4) that
Similarly,
Therefore, by the above inequality
when, we get
In view of (H4), there exist constant, and let, such that
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.
3.2. Global Attractor
Theorem 3.2. [11] 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
2) It exists a bounded absorbing set, namely, , it exists a constant, so that
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
where
is the bounded absorbing set of and satisfies.
1);
2), here and it is a bounded set,
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,
This shows that is uniformly bounded in.
2) Furthermore, for any, when, we have
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.
Cite this paper
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
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. 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.