The Global Attractor of Thermoelastic Coupled System *

In this paper, we consider a class of Sine-Gordon equations which arise from the model of the thermoelastic coupled rod. Firstly, by virtue of the classical semigroup theory, we prove the existence and uniqueness of the mild solution under certain initial-boundary value for above-mentioned equations. Secondly, we obtain the boundedness of solutions by the priori estimates. Lastly, we prove the existence of a global attractor.


Introduction
In this paper, we consider the following nonlinear thermoelastic coupled rod system The above system describes the vibrations of an extensible thermoelastic rod model.Here u = u(x, t) and θ = θ(x, t) are all real-valued functions on Ω × [0, +∞], Ω = (0, 1) is an open bounded domain of R. The coefficient α, γ, ε, k are all positive constants, where α is called the strong damping coefficient of rod, ε is the small parameter.The sign Δ denotes ,      "Global solutions" and "global attractor" are two basic concepts in the study of long-time behavior of nonlinear dissipative evolution equations with various dissipation.If the coupled terms are equivalent to 0, Equations (1) will decouple to the Sine-Gordon equation and the heat equation.The structure of global attractors for weakly damped nonlinear wave Equation (2) as α, ε = 0 is studied in Temam [1] and Zhu [2] and Wang [3] and the one for the strongly damped nonlinear wave equation is considered in Zhou [4].Semion [5] shows the Frechet differentiability for a damped sine-Gordon equation with a variable diffusion coefficient.Han [6] proves the existence of Random attractors for stochastic Sine-Gordon lattice system.But have the global solution and the global attractor for the "thermoelastic coupled" rod system (1)?To our knowledge, nothing was known until now.
In this paper, we give the proof of the existence and uniqueness of the mild solution and the existence of a global attractor for system (1)

Existence and Uniqueness of Global Solutions
It is well known that operator

positive and linear and its eigenvalues
 with the usual inner products and norms, respectively It is convenient to reduce (1) to an evolution equation of the first order in time, Let u t = v, then (1) are equivalent to the following initial value problem in E, H  and D(A).Set B = -C, then similar to [7], by ma e slight modification and reasoning, we can prove that for any α, k > 0, B is a sectorial operator on E and generates an analytic semigroup Ct e on E for t > 0. By the assumptions g 1 (x), , it is easy to check that the function F(Y, lly Lipschitz continuous with respect to Y, By the classical semigroup theory concerning the existence and uniqueness of the solution of evolution differential equations, we have Theorem Assume that the assumptions g (x), d In this case, Y(t) is called a mild solution of the system (3 x H   hold, then consider the initial value Hilbert space E. For any initial value ; is co oduce existence of the bounded absorbing set and the global Absorbing Set a ntinuous in E. In the following, we will intr the attractor for map {S(t), t > 0} in E.

The Existence of Bounded
In this section, we will show boundedness of the solu tions for system (3).For this purpose, we define weighted inner product and norm in , for any  , where Obviously the norm then the system (3) can be written as where Obviously, the mapping , (6) has the relation with where R ε {u, w, θ} →{u, w + εu, θ} is an omorphism of E. So we only need consider the equivale map (6).
po is nt For the boundedness of the solution of ( 5) in E, we firstly present an important lemma which plays an im rtant role in this artical.Lemma 3.1.For any where , the Lemma is easily obtained.Theorem 3.1.There exists a positive co t M > 0 such that for any bounded set of E, then there exists T 0 (B) ≥ 0 such that the solution φ = (u(t), w(t), θ(t)) T of ( 5) , , By Gronwall's inequality we have thus we complete the proof of the Theorem.Corollary 3.1.Let B 0 be a bounded closed ball of E at 0 of radius M. For any in alue centered itial v , there exists a constant M 1 = M 1 (M) such t)) T satisfies that the solution of ( 5) tence of a Global Attractor
By putting x = 0, 1 and t = 0 in (10) we get , ( 7) is easily The operator 1 2 A operates into (10) to get Taking the inner product of ( 14) with , , d Considering to (13) we ge