IJMNTAInternational Journal of Modern Nonlinear Theory and Application2167-9479Scientific Research Publishing10.4236/ijmnta.2016.51008IJMNTA-64631ArticlesEngineering Physics&Mathematics Global Attractors for a Class of Generalized Nonlinear Kirchhoff-Sine-Gordon Equation uijinLou1PenghuiLv1GuoguangLin1*Mathematical of Yunnan University, Kunming, China* E-mail:15925159599@163.com(GL);020320160501738129 December 2015accepted 14 March 17 March 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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.

Kirchhoff-Sine-Gordon Equation The Existence and Uniqueness of Solutions Priori Estimates Global Attractors
1. Introduction

In 1883, Kirchhoff  proposed the following model in the study of elastic string free vibration:

, where is associated with the initial tension, M is related to the material

properties of the rope, and indicates the vertical displacement at the x point on the t. The equation is more accurate than the classical wave equation to describe the motion of an elastic rod.

Masamro  proposed the Kirchhoff equation with dissipation and damping term:

where is a bounded domain of with a smooth boundary; he uses the Galerkin method to prove the existence of the solution of the equation at the initial boundary conditions.

Sine-Gordon equation is a very useful model in physics. In 1962, Josephson  fist applied the Sine-Gordon equation to superconductors, where the equation:, is the two-order partial derivative of u with respect to the variable t; is the two-order partial derivative of the u about the independent variable x. Subsequently, Zhu  considered the following problem: (where is a bounded domain of) and he proved the existence of the global solution of the equation. For more research on the global solutions and global attractors of Kirchhoff and sine-Gordon equations, we refer the reader to  - .

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

where is a bounded domain of with a smooth boundary; is the dissipation coefficient; is a positive constant; and is the external interference. The assumptions on nonlinear terms and will be specified later.

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.

2. Basic Assumption

For brevity, we define the Sobolev space as follows:

In addition, we define and are the inner product and norm of H.

Nonlinear function satisfying condition (G):

Function satisfies the condition (F):

3. A Priori Estimates

Lemma 3.1. Assuming the nonlinear function satisfies the condition (G)-(F), , , then the solution of the initial boundary value problem (1.1) satisfies and

where. Thus there exists a positive constant and, such that

Proof. Let, the equation can be transformed into

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 is the first eigenvalue of with Dirichlet boundary conditions on.

Since and (F) (6), (7), we get

and

where

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

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

, and since

that is

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

Set, and, then (3.11) is equivalent to (3.12)

where

By using Gronwall inequality, we obtain

Let.

So, we have

then

Hence, there exists and, such that

Lemma 3.2. Assuming the nonlinear function satisfies the condition (G)-(F), , then the solution of satisfies the initial boundary value problem (1.1) satisfies and

where. Thus there exists a positive constant and, such that

Proof. The equations (3.1) in the H and have inner product, we find that

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, this will imply

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

Let, and, then (3.25) type can be changed into

then

where.

By using Gronwall inequality, we obtain

taking, we have

then

Hence, there exists and, such that

Theorem 3.1. Assuming the nonlinear function satisfies the condition (G)-(F), , , so the initial boundary value problem (1.1) exists a unique smooth solution.

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

ation, the proof procedure is omitted. Next, we prove the uniqueness of solutions in

detail.

Assume are two solutions of equation, we denote, then, the two equations subtract and obtain

We take the inner product of the above equations (3.31) with in H, we have

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, then (3.38) can be changed to

By using Gronwall inequality, we obtain

There has

That show that.

So as to get, the uniqueness is proved. ■

4. Global Attractor

Theorem 4.1.  Set be a Banach space, and are the semigroup operator on.; here I is a unit operator. Set satisfy the follow conditions.

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

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

here and B are bounded sets.

3) When, is a completely continuous operator.

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

Theorem 4.2.  Under the assume of 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 3.1-Lemma 3.2, 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) is completely continuous. ■

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

Acknowledgements

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.

Funding

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11161057.

Cite this paper

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

ReferencesKirchhof, G. (1883) Vorlesungen fiber Mechanik. Teubner, Stuttgarty.Masamro, H. and Yoshio, Y. (1991) On Some Nonlinear Wave Equations 2: Global Existence and Energy Decay of Solutions. J. Fac. Sci. Univ. Tokyo. Sect. IA, Math., 38, 239-250.Josephson, B.D. (1962) Possible New Effects in Superconductive Tunneling. Physics Letters, 1, 251-253. http://dx.doi.org/10.1016/0031-9163(62)91369-0Zhu, Z.W. and Lu, Y. (2000) The Existence and Uniqueness of Solution for Generalized Sine-Gordon Equation. Chinese Quarterly Journal of Mathematics, 15, 71-77.Li, Q.X. and Zhong, T. (2002) Existence of Global Solutions for Kirchhoff Type Equations with Dissipation and Damping Terms. Journal of Xiamen University: Natural Science Edition, 41, 419-422.Silva, M.A.J. and Ma, T.F. (2013) Long-Time Dynamics for a Class of Kirchhoff Models with Memory. Journal of Mathematical Physics, 54, Article ID: 021505.Zhang, J.W., Wang, D.X. and Wu, R.H. (2008) Global Solutions for a Class of Generalized Strongly Damped Sine-Gordon Equation. Journal of Mathematical Physics, 57, 2021-2025.Guo, L., Yuan, Z.Q. and Lin, G.G. (2014) The Global Attractors for a Nonlinear Viscoelastic Wave Equation with Strong Damping and Linear Damping and Source Terms. International Journal of Modern Nonlinear Theory and Application, 4, 142-152. http://dx.doi.org/10.4236/ijmnta.2015.42010Teman, R. (1988) Infiniter-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York, 15-26. http://dx.doi.org/10.1007/978-1-4684-0313-8_2Ma, Q.F., Wang, S.H. and Zhong, C.K. (2002) Necessary and Sufficient Congitions for the Existence of Global Attractors for Semigroup and Applications. Indiana University Mathematics Journal, 51, 1541-1559. http://dx.doi.org/10.1512/iumj.2002.51.2255Ma, Q.Z., Sun, C.Y. and Zhong, C.K. (2007) The Existence of Strong Global Attractors for Nonlinear Beam Equations. Journal of Mathematical Physics, 27A, 941-948.Lin, G.G. (2011) Nonlinear Evolution Equation. Yunnan University Press, 12.