Stochastic Nonlinear Thermoelastic System Coupled Sine-Gordon Equation Driven by Jump Noise

and Applied Analysis 3 3. Existence and Uniqueness of Solution In this section, we use the Galerkin method to prove the local existence and uniqueness of solution; then making use of a priori estimates, we prove that there exists a convergence subsequence such that the solution is global. As is well known, system (1) is equivalent to the following Itô system: du = Vdt, dV = (α 1 V xx + u xx + β sin u + α 2 θ) dt + ∫ Z σ 1 (t, u (t) , z) η 1 (dz, dt) , dθ = (θ xx − α 2 V + g (t, θ)) dt + ∫ Z σ 2 (t, θ (t) , z) η 2 (dz, dt) , u (x, t) = 0, θ (x, t) = 0, x = 0, x = L, t ≥ 0, u (x, 0) = u 0 (x) , V (x, 0) = u 1 (x) , θ (x, 0) = θ 0 (x) , x ∈ [0, L] . (10) For simplicity, denote that g(θ) := g(t, θ), σ i (u) := σ i (t, u, z) (i = 1, 2), and L2(0, L) = L, Hk(0, L) = Hk, k ∈ N . To obtain the existence of solution to (10), we suppose that the functions σ i : [0,∞) × L 2 (0, L) 󳨀→ L 2 (Z, ]; L2 (0, L)) , g : [0,∞) × L 2 (0, L) 󳨀→ L 2 (0, L) (11) satisfy the following conditions: C 1 : 󵄩󵄩󵄩󵄩σi(u) 󵄩󵄩󵄩󵄩 2 L 2 (Z,];L(0,L)) ≤ k1‖u‖ 2 , 󵄩󵄩󵄩󵄩g(u) 󵄩󵄩󵄩󵄩 2 ≤ K 1 ‖u‖ 2 , C 2 : 󵄩󵄩󵄩󵄩σi (u) − σi (V) 󵄩󵄩󵄩󵄩 2 L (Z,];L(0,L)) ≤ k2‖u − V‖ 2 , 󵄩󵄩󵄩󵄩g(u) − g(V) 󵄩󵄩󵄩󵄩 2 ≤ K 2 ‖u − V‖2, (12) where k i , K i > 0, i = 1, 2 and u, V ∈ L2(0, L). Definition 2. An F t -adapted stochastic process {(u(t), V(t), θ(t))} t≥0 is said to be a strong probabilistic solution of stochastic nonlinear thermoelastic coupled system driven by Lévy noise (10) if it satisfies the following: (1) (u(t), V(t), θ(t)) ∈ L2(Ω; C([0, T];H1 × L2 × L2)) a.s. for any T > 0; (2) the identities (u (t) , φ 1 ) = (u 0 , φ 1 ) + ∫ t 0 (V (s) , φ 1 ) ds, (V (t) , φ 2 ) = (V 0 , φ 2 )

Recently, the study of high-temperature apparatus or heat resistant structures is becoming important and it is necessary to analyze not only the deterministic thermal stress but also the stochastic thermal stress.In high-temperature apparatus, it is very difficult to predict accurately the thermal environment and mechanical load on its components.Furthermore, many indeterminate factors must be considered, for example, the random high-cycle vibrations of the temperature of the upper core in fast breeder reactors and fluctuations in the heat transfer coefficients around the stationary blades of gas turbines.Therefore, the stochastic case of temperature and thermal stress is indispensable in considering these indeterminate factors of the thermal environment (see [5]).
All the time, a description of wave propagation phenomena in random media is usually based on the study of stochastically perturbed wave equations (see [6,7]).In fact, lots of wave phenomena are temperature dependent or heat generating; then the wave equations are coupled with a stochastic heat equation.Caraballo et al. [8] studied the existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations.Bates et al. [9] proved the existence of random attractors for stochastic reaction-diffusion equations on unbounded domains, and Wang and Tang [10,11] described the properties of the random attractors.
Meanwhile the sine-Gordon equation is an important model in physics; Fan [12] considered the random attractor for the stochastic sine-Gordon equation.What are the other properties of stochastic sine-Gordon equation?As we know, Coayla-Teran [13] and Liu et al. [3] studied the mild solution
Let (, Z) be a measurable space, and let ] be a -finite positive measure on it.If  is a topological space, then by B() we will denote the Borel -field on , and  is a Lebesgue measure on (, B()).Suppose that (Ω, F, , ) is a filtered probability space, where  = (F  ) ≥0 is a filtration and   : Ω×B( + )×Z →  is a time homogeneous Poisson random measure with the intensity measure ] defined over the filtered probability space (Ω, F, , ).
We will denote by η =   −   the compensated Poisson random measure associated with   , where the compensator   is given by We assume that (, | ⋅ |  ) is a Hilbert space.It is then known (see, for example, [1,2]) that there exists a unique continuous linear operator I which associates with each progressively measurable process  : Moreover, I() is an -valued adapted and càdlàg process such that for any random step process (, ) satisfying the condition (3) with a representation where and for all ,   being an F  −1 measurable random variable, one has In general case we write The continuity (more precisely, isometry in Hilbert spaces) of the operator I mentioned above means that The class of all progressively measurable processes  :  + ×  × Ω →  satisfying the condition (3) will be denoted by M 2 ( + ,  2 (, ], )).If  > 0, the class of all progressively measurable processes  : [0, ] ×  × Ω →  satisfies the condition (3) just for this one , which will be denoted by M 2 (0, ,  2 (, ], )).
The main technical tool in our paper is the Itô formula.Let us consider the Hilbert spaces  ⊆  ≅   ⊆   and a   -valued càdlàg process of the form where  is a   -valued process and  is an -valued process; we have the following.

Existence and Uniqueness of Solution
In this section, we use the Galerkin method to prove the local existence and uniqueness of solution; then making use of a priori estimates, we prove that there exists a convergence subsequence such that the solution is global.
We now give our main result.
where  is a positive constant.
Remark 5. Since (, , ) denotes the displacement at point (, ) on an orbit  ∈ Ω, a.s., / = V means the velocity; the result in the Theorem 4 exhibits that the velocity is exponentially decay in time  in the sense of mean square; in the view of physics, one can obtain that the displacement  will tend to a constant in the large time in the sense of mean square.