Global Attractors for a Class of Generalized Nonlinear Kirchhoff-Sine-Gordon Equation

In this paper, we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation ( ) ( ) ( ) tt t t u u u u u g u f x 2 sin β α φ − ∆ + − ∇ ∆ + = . 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.


Introduction
In 1883, Kirchhoff [1] proposed the following model in the study of elastic string free vibration: , where α is associated with the initial tension, M is related to the material properties of the rope, and ( ) , u x t 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 [2] proposed the Kirchhoff equation with dissipation and damping term: , 0 , 0 , 0 , 0 , , 0 where Ω is a bounded domain of ( ) R n ≥ with a smooth boundary ∂Ω ; 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 [3] fist applied the Sine-Gordon equation to superconductors, where the equation: sin 0 tt xx u u u − + = , tt u is the two-order partial derivative of u with respect to the variable t; xx u is the two-order partial derivative of the u about the independent variable x.Subsequently, Zhu [4] considered the following problem: where Ω is a bounded domain of 3  R ) 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 [5]- [11].
Based on Kirchhoff and Sine-Gordon model, we study the following initial boundary value problem: where Ω is a bounded domain of ( ) with a smooth boundary ∂Ω ; α is the dissipation coefficient; β is a positive constant; and ( ) f x is the external interference.The assumptions on nonlinear terms ( ) 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.

Basic Assumption
For brevity, we define the Sobolev space as follows: In addition, we define ( ) • • and • are the inner product and norm of H.

A Priori Estimates
, then the solution ( ) . Thus there exists a positive constant ( ) 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
, then the solution ( ) . Thus there exists a positive constant ( ) , .
Proof.The equations (3.1) in the H and By using Holder inequality, Young's inequality and Poincare inequality, we get the following results
Hence, there exists ( ) , .( ) , 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- , the proof procedure is omitted.Next, we prove the uniqueness of solutions in detail.
Assume , u v are two solutions of equation, we denote w u v = − , then, the two equations subtract and ob- tain