Global Existence of the Higher-Dimensional Linear System of Thermoviscoelasticity

We obtain a global existence result for the higher-dimensional thermoviscoelastic equations. Using semigroup approach, we will establish the global existence of homogeneous, nonhomogeneous, linear, semilinear, and nonlinear, thermoviscoelastic systems.


Introduction
In this paper, we consider global existence of the following thermoviscoelastic model: where the sign " * " denotes the convolution product in time, which is defined by The body Ω is a bounded domain in R n with smooth boundary Γ ∂Ω say C 2 and is assumed to be linear, homogeneous, and isotropic. u x, t u 1 x, t , u 2 x, t , . . . , u n x, t , and θ x, t represent displacement vector and temperature derivations, respectively, from the natural state of the reference configuration at position x and time t. λ, μ > 0 are Lamé's constants and α, β > 0 the coupling parameters; g t denotes the relaxation function, w 0 x, s is a specified "history," and u 0 x , u 1 x , θ 0 x are initial data. Δ, ∇, div denote the Laplace, gradient, and divergence operators in the space variables, respectively.

1.6
In the one-dimensional space case, there are many works see e.g., 4-8 on the global existence and uniqueness. Liu and Zheng 9 succeeded in deriving in energy decay under the boundary condition 1.4 or or u| x 0 0, σ| x l 0, θ| x 0, l 0, 1.9 and Hansen 10 used the method of combining the Fourier series expansion with decoupling technique to solve the exponential stability under the following boundary condition: International Journal of Differential Equations   3 where σ u x − γθ is the stress. Zhang and Zuazua 11 studied the decay of energy for the problem of the linear thermoelastic system of type III by using the classical energy method and the spectral method, and they obtained the exponential stability in one space dimension, and in two or three space dimensions for radially symmetric situations while the energy decays polynomially for most domains in two space dimensions. When α β 0, f h 0, system 1.1 -1.4 is decoupled into the following viscoelastic system: and the wave equation.
There are many works see, e.g., 9, 12-15 on exponential stability of energy and asymptotic stability of solution under different assumptions. The notation in this paper will be as follows.
0 denote the usual Sobolev spaces on Ω. In addition, · B denotes the norm in the space B; we also put · · L 2 Ω . We denote by C k I, B , k ∈ N 0 , the space of k-times continuously differentiable functions from J ⊆ I into a Banach space B, and likewise by L p I, B , 1 ≤ p ≤ ∞, the corresponding Lebesgue spaces. C β 0, T , B denotes the Hölder space of B-valued continuous functions with exponent β ∈ 0, 1 in variable t.

Main Results
Let the "history space" L 2 g, 0, ∞ , H 1 0 Ω n consist of H 1 0 Ω n -valued functions w on 0, ∞ for which where κ denotes the positive constant in H 3 , that is,

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Thus we consider the following thermoviscoelastic system:

2.7
System 2.5 can be written as follows:

2.8
We define a linear unbounded operator A on H by International Journal of Differential Equations 5 where w s ∂w/∂s and B u, w κμΔu κ λ μ ∇ div u μ ∞ 0 g s Δw s ds λ μ ∞ 0 g s ∇ div w s ds.

2.11
Then problem 2.8 can be formulated as an abstract Cauchy problem where

It is clear that D A is dense in H.
Our hypotheses on f, h can be stated as follows, which will be used in different theorems: We are now in a position to state our main theorems.

Theorem 2.2. Suppose that condition (A 2 ) holds. Relaxation function g satisfies (H 1 )-( H 3 ). Then for any
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is a nonlinear operator from D A into D A and satisfies the global Lipschitz condition on
Then for any Φ 0 International Journal of Differential Equations 7

Some Lemmas
In this section in order to complete proofs of Theorems 2.1-2.6, we need first Lemmas 3.1-3.5.
For the abstract initial value problem, where B is a maximal accretive operator defined in a dense subset D B of a Banach space H.
We have the following. Proof. We first prove the necessity. B is an accretive operator, so we have Thus, for all λ > 0, Letting λ → 0, we get i . Furthermore, ii immediately follows from the fact that B is maccretive. We now prove the sufficiency. It follows from i that for all λ > 0,

3.4
Now it remains to prove that B is densely defined. We use a contradiction argument. Suppose that it is not true. Then there is a nontrivial element x 0 belonging to orthogonal supplement of D B such that for all x ∈ D B , x, x 0 0.

3.5
It follows from ii that there is x * ∈ D B such that Taking the inner product of 3.5 with x * , we deduce that International Journal of Differential Equations Taking the real part of 3.7 , we deduce that x * 0, and by 3.6 , x 0 0, which is a contradiction. Thus the proof is complete.

Lemma 3.2.
Suppose that B is m-accretive in a Banach space H, and u 0 ∈ D B . Then problem 3.1 has a unique classical solution u such that 3.8

Lemma 3.3. Suppose that K K t , and
Then problem 3.1 admits a unique global classical solution u such that which can be expressed as Proof. Since S t u 0 satisfies the homogeneous equation and nonhomogeneous initial condition, it suffices to verify that w t given by

3.13
When h → 0, the terms in the last line of 3.13 have limits: 3.14 International Journal of Differential Equations 9 It turns out that w ∈ C 1 0, ∞ , H and the terms in the third line of 3.13 have limits too, which should be Thus the proof is complete.

3.16
Then problem 3.1 admits a unique global classical solution.
Proof. From the proof of Lemma 3.2, we can obtain

3.17
When h → 0, the last terms in the line of 3.17 have limits:

3.18
Combining the results of Lemma 3.3 proves the lemma.

Lemma 3.5. Suppose that K K t , and
and for any T > 0,

3.20
Then problem 3.1 admits a unique global classical solution.
Proof. We first prove that for any K 1 ∈ L 1 0, T , H , the function w given by the following integral: belongs to C 0, T , H . Indeed, we infer from the difference where we have used the strong continuity of S t and the absolute continuity of integral for K 1 ∈ L 1 0, t . Now it can be seen from the last line of 3.13 that for almost every t ∈ 0, T , dw/dt exists, and it equals

3.24
Thus, for almost every t,

3.25
Since w and K both belong to C 0, T , H , it follows from 3.25 that for almost every t, Bw equals a function belonging to C 0, T , H . Since B is a closed operator, we conclude that w ∈ C 0, T , D B ∩ C 1 0, T , H 3.26 and 3.25 holds for every t. Thus the proof is complete.
To prove that the operator A defined by 2.14 is dissipative, we need the following lemma.

3.28
Proof. See, for example, the work by Liu in 16 . Proof. By a straightforward calculation, it follows from Lemma 3.7 that

3.29
Thus, A is dissipative.

3.35
Inserting v g 1 and θ t g 3 obtained from 3.31 , 3.33 into 3.34 , we obtain By the standard theory for the linear elliptic equations, we have a unique θ ∈ H 2 Ω ∩ H 1 0 Ω satisfying 3.36 .
We plug v g 1 obtained from 3.31 into 3.35 to get

3.37
Applying the standard theory for the linear elliptic equations again, we have a unique w ∈ H 1 g, 0, ∞ , H 1 0 Ω n satisfying 3.37 . Then plugging θ and w just obtained from solving 3.36 , 3.37 , respectively, into 3.32 and applying the standard theory for the linear elliptic equations again yield the unique solvability of u ∈ D A for 3.32 and such that κu ∞ 0 g s w s ds ∈ H 2 Ω ∩ H 1 0 Ω n . Thus the unique solvability of 3.30 follows. It is clear from the regularity theory for the linear elliptic equations that Φ H ≤ K G H with K being a positive constant independent of Φ. Thus the proof is completed. Lemma 3.9. The operator A defined by 2.13 is closed.
Proof. To prove that A is closed, let u n , v n , θ n , θ nt , w n ∈ D A be such that u n , v n , θ n , θ nt , w n −→ u, v, θ, θ t , w in H, A u n , v n , θ n , θ tn , w n −→ a, b, c, d, e in H.

3.38
Then we have

3.50
By 3.42 and 3.46 , we deduce International Journal of Differential Equations 13 By 3.47 and 3.49 , we deduce and consequently, it follows from 3.41 , that since Δ is an isomorphism from H 2 Ω ∩ H 1 0 Ω onto L 2 Ω . It therefore follows from 3.47 and 3.54 that By 3.43 , 3.48 , and 3.49 , we deduce Proof of Corollary 2.3. By A 3 or A 4 , we derive that K 0, f, 0, h, 0 ∈ C 0, ∞ , D A or K ∈ C 0, ∞ , H , and for any T > 0, K t ∈ L 1 0, T , H . Noting that B −A is the maximal accretive operator, we use Lemmas 3.4 and 3.5 to prove the corollary.
Proof of Corollary 2.4. We know that K x, t 0, f, 0, h, 0 are Lipschitz continuous functions from 0, T into H. Moreover, by 2.2 , it is clear that H is a reflexive Banach space. Therefore, K t ∈ L 1 0, T , H . Hence applying Lemma 3.5, we may complete the proof of the corollary.
Proof of Theorem 2.5. By virtue of the proof of Theorem 2.2, we know that B −A is the maximal accretive operator of a C 0 semigroup S t . On the other hand, K 0, f, 0, h, 0 satisfies the global Lipschitz condition on H. Therefore, we use the contraction mapping theorem to prove the present theorem. Two key steps for using the contraction mapping theorem are to figure out a closed set of the considered Banach space and an auxiliary problem so that the nonlinear operator defined by the auxiliary problem maps from this closed set into itself and turns out to be a contraction. In the following we proceed along this line. Let where k is a positive constant such that k > L. In Ω, we introduce the following norm: Clearly, Ω is a Banach space. We now show that the nonlinear operator φ defined by 4.1 maps Ω into itself, and the mapping is a contraction. Indeed, for Φ ∈ Ω, we have

4.6
Therefore, by the contraction mapping theorem, the problem has a unique solution in Ω.
To show that the uniqueness also holds in C 0, ∞ , H , let Φ 1 , Φ 2 ∈ C 0, ∞ , H be two solutions of the problem and let Φ Φ 1 − Φ 2 . Then

4.7
By the Gronwall inequality, we immediately conclude that Φ t 0; that is, the uniqueness in C 0, ∞ , H follows. Thus the proof is complete.

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Proof of Theorem 2.6. Since B is the maximal accretive operator, K 0, f, 0, h, 0 satisfies the global Lipschitz condition on D A . Let Then A 1 is a Banach space, and B 1 B 2 is a densely defined operator from D B 2 into A 1 . In what follows we prove that B 1 is m-accretive in A 1 D B . Indeed, for any x, y ∈ D B 2 , since B is accretive in H, we have x − y λ Bx − By D B x − y λ Bx − By This implies that S 1 t is a restriction of S t on A 1 . By virtue of the proof of Theorem 2.5, there exists a unique mild solution Φ ∈ C 0, ∞ , A 1 . Since S 1 t is a restriction of S t on D B , and moreover, we infer from K Φ being an operator from D B to D B and Lemma 3.4 that Φ is a classical solution to the problem. Thus the proof is complete.