1. Introduction
In this paper, we consider the following nonlinear thermoelastic coupled rod system
(1)
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
(2)
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) in space 
× L2(Ω) × L2(Ω).
2. Existence and Uniqueness of Global Solutions
It is well known that operator A = –Δ:D(A) = 
H2(Ω) → L2(Ω) is self-adjoint, positive and linear and its eigenvalues 
satisfy 0 < λ1 ≤ λ2 ≤ ··· ≤ λm ≤ ··· and λm → +∞ as m → +∞. Set L2(Ω), 
, 
, 
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 ut = v, then (1) are equivalent to the following initial value problem in E,
(3)
where Y = (u, v, θ)T,
with D(C) = D(A) × D(A) × D(A), I is the identity operator L2(Ω), 
and D(A). Set B = –C, then similar to [7], by making some 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 
on E for t > 0. By the assumptions g1(x), 
, it is easy to check that the function F(Y, t):E → E is locally 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 2.1. Assume that the assumptions g1(x), 
hold, then consider the initial value problem (3) in Hilbert space E. For any initial value
, there exists a unique continuous function Y(t) = Y(t, Y0) = 
such that Y(t) = (u(t), v(t), θ(t))T satisfies the integral equation
.
In this case, Y(t) is called a mild solution of the system (3) and Y(t) = Y(t, Y0) is jointly continuous in t and Y0, that is, the solution (u, ut, θ) of the system (1) satisfies
.
By Theorem 2.1, for any t > 0, we may introduce the map
It maps E = 
× L2(Ω) × L2(Ω) into itself and it enjoys the usual process properties as follows
It is obviously that the map {S(t), t > 0}, 
is continuous in E. In the following, we will introduce the existence of the bounded absorbing set and the global attractor for map {S(t), t > 0} in E.
3. The Existence of Bounded Absorbing Set
In this section, we will show boundedness of the solutions for system (3). For this purpose, we define a weighted inner product and norm in E = 
× L2(Ω) × L2(Ω) by
(4)
for any
, where wi = uit + εi, i = 1, 2 and μ is chosen by
.
Obviously the norm 
in (4) is equivalent to the usual norm in space E.
Let φ = (u, w, θ)T, w = ut + εu, where ε is chosen as
then the system (3) can be written as
(5)
where 
and 
Obviously, the mapping
(6)
has the relation with
where Rε{u, w, θ} →{u, w + εu, θ} is an isomorphism of E. So we only need consider the equivalent map (6).
For the boundedness of the solution of (5) in E, we firstly present an important lemma which plays an important role in this artical.
Lemma 3.1. For any 
× L2(Ω) × L2(Ω). we have
.
where
are all positive constants if
.
Proof.
with
the Lemma is easily obtained.
Theorem 3.1. There exists a positive constant M > 0 such that for any bounded set B of E, then there exists T0(B) ≥ 0 such that the solution φ = (u(t), w(t), θ(t))T of (5) with 
satisfies
in which w = ut + εu.
Proof. Let φ = (u, w, θ)T, w = ut + εu be the solution of (5) with initial value
. Taking the inner product 
of (5) with φ = (u, w, θ)T and considering of Lemma 3.1, we have
with
we have
By Gronwall’s inequality we have
Set 
thus we complete the proof of the Theorem.
Corollary 3.1. Let B0 be a bounded closed ball of E centered at 0 of radius M. For any initial value
, there exists a constant M1 = M1(M) such that the solution of (5) φ(t) = (u(t), w(t), θ(t))T satisfies
.
4. The Existence of a Global Attractor
Theorem 4.1. For any initial value
, the solution φ(t) = (u(t), w(t), θ(t))T of (5) can be decomposed into φ(t) = φ1(t) + φ2(t) where φi(t) = (ui(t), wi(t), θi(t))T, wi(t) = uit(t) + εui(t), i = 1, 2 satisfy, respectively(
)
(7)
(8)
where 
and 
depend on M.
Proof. Let φ(t) = (u(t), w(t), θ(t))T, 
be a solution of (5) in space E with the initial value
.
Let φ(t) =φ1(t) +φ2(t), where φi(t) = (ui(t), wi(t), θi(t))T, wi(t) = uit(t) + εui(t), i = 1, 2, satisfy, respectively
(9)
where
and
(10)
where
Taking the inner product of (9) with φ1(t) we have
(11)
Similar to the proof of Lemma 3.1, we can obtain
(12)
where 
.
Thus by (12) and Gronwall’s inequality, (7) is easily obtained from (11).
By putting x = 0, 1 and t = 0 in (10) we get
(13)
The operator 
operates into (10) to get
(14)
Taking the inner product of (14) with 
(15)
Considering to (13) we get that
with
from (15) we have
(16)
Similar to the proof of the Lemma 3.1, from (16) we get
By the Gronwall’s inequality, we have
Set
.
Thus, the proof of Theorem 4.1 is complete.
NOTES
#Corresponding author.