ON GLOBAL SOLUTIONS IN ONE-DIMENSIONAL THERMOELASTICITY WITH SECOND SOUND IN THE HALF LINE

. In this paper, we investigate the initial boundary value problem for one-dimensional thermoelasticity with second sound in the half line. By using delicate energy estimates, together with a special form of Helmholtz free energy, we are able to show the global solutions exist under the Dirichlet boundary condition if the initial data are suﬃcient small.

Here, for simplicity, we assume that the relaxation parameter τ is a positive constant and the heat conductive coefficient κ is also a given constant. Note that the Clausius-Duhem inequality ψ t + ηθ t − Sω t + qθ x θ + T 0 ≤ 0 1672 YUXI HU AND NA WANG holds under this special choice of the Helmholtz free energy. Therefore, in view of (2) and (3), the original system (1) can be written as where a(ω, θ) =ψ ωω , b(ω, θ) = −ψ ωθ ,ã(ω, θ, q) = −ψ θθ , We consider the initial boundary value problem for the functions ω, v, θ, q : Ω × [0, ∞) −→ R with initial conditions (ω(x, 0), v(x, 0), θ(x, 0), q(x, 0)) = (ω 0 (x), v 0 (x), θ 0 (x), q 0 (x)), and Dirichlet boundary conditions (rigidly clamped, constant temperature) where Ω = (0, +∞) and the initial data v 0 , θ 0 should be compatible with the boundary condition. For the limit case τ = 0, the equations (1) and (2) constitute the system of one-dimensional classical thermoelasticity, in which the relation between the heat flux and the temperature is governed by Fourier's law, For this system, global existence of solutions for small data has been established for various domains, i.e., the whole space, the half space and the bounded interval, see [11,19,20,12,13,14]. The formation of singularities for large data was considered by Dafermos and Hsiao [4], Hrusa and Messaoudi [10], and Hansen [6]. For more results in this respect, see the monograph [15] for details. In case τ > 0, Fourier's law is replaced by Cattaneo's law (4) 4 for heat conduction, and the corresponding system is called thermoelasticity with second sound reflecting the appearance of heat waves with finite propagation speed. For the Cauchy problem, Tarabek [26] has established a global existence theorem for small initial data, and the decay to an equilibrium. The local existence theorem, stated in [26] with a hint to the paper of Hughes, Kato and Marsden [9], was completely proved, and the decay rates for global solutions were given by Racke and Wang [23]. Recently, by using weighted energy estimates, A. Kasimov, R. Racke and B. Said-Houari [16] proved a global existence theorem for small data with improved decay rates. In [16], they first analyzed the decay property for linearized model by using the classical Shizuta-Kawashima condition [25] and then proved the decay estimates for both solutions and its derivatives of the nonlinear system. For initial boundary value problem, Racke [21] has proved the exponential stability and global existence on bounded domain, see [18] for the general case. In the case Ω = R + , Hu [7] proved the global existence for small data under the traction free and constant temperature boundary condition, i.e. ω| ∂Ω = θ| ∂Ω = 0.
Since the Poincaré inequality is invalid for unbounded domains, the methods used for bounded domains cannot be carried over into unbounded case. This is the main difficulty for the half line problem. The main point in [7] is that the special type of boundary condition allows one to do Fourier analysis which helps to get the decay rates of the linearized system and thus establish the well-posedness of the nonlinear system. However, for Dirichlet boundary problem, the Fourier analysis fails. We shall use other methods to solve this problem.
This paper is mainly motivated by Jiang's paper [14] where he obtained the global existence of smooth solutions to the system of classical thermoelasticity under Dirichlet boundary condition in the half line by directly using energy estimates. We observe that for a special type of Helmholtz free energy, we can get the lower energy estimates of system (4) immediately, which makes it possible to do the higher order estimates. This type of Helmholtz free energy is physical reasonable for second sound (see [26]) and was also used in dealing with the Cauchy problem in [26]. Moreover, since the classical thermoelasticity have more dissipation than the second sound case, one must be careful to deal with the boundary terms which is quite different compared with that in [14]. We adapted the method used in [21] to overcome this difficulties and finally obtain the desired a priori estimates. We note that even though there are many results in classical thermoelasticity that can be extended to thermoelasticity with second sound, see [19,21] and [23], it is not true, for example, for Timoshenko-type thermoelastic systems, where a system can be or remain exponentially stable under Fourier's law, while it loses this property under Cattaneo's law, see [5]. Therefore, our results are meaningful in this respect.
We note that a global defined smooth solution for thermoelasticity with second sound should not be expected for large data. Indeed, Hu and Racke [8] showed that for special choice of constitutive equations, the solution to the Cauchy problem will develop singularities in finite time if the initial data are large.
We now introduce some notations which will be frequently used throughout the paper. For a non-negative integer N, let We denote by W m,p (Ω), 0 ≤ m ≤ ∞, 1 ≤ p ≤ ∞, the usual Sobolev space with the norm · W m,p . For convenience, H m (Ω) and L p (Ω) stand for W m,2 (Ω) and W 0,p (Ω) with the norm · H m and · L p , respectively. For p = 2, we denote the norm · L 2 by · . Let X be a Banach space. We denote by L p ([α, β], X) (1 ≤ p ≤ ∞) and · L p ([α,β],X) the space of all measurable p-th power functions from [α, β] to X and its norm, respectively.
For U = (ω, v, θ, q) a function of t and x, we denote In order to formulate the compatibility conditions for the initial data U 0 ( We shall suppose throughout the paper that: . There exists positive constants γ 0 , γ 1 such that (10).
Here we present the main theorem and will give a proof in the end of this paper.

Remark 2.
Our system is a partially dissipative quasi-linear hyperbolic system, see (14) below. We mention that for Cauchy problem of such systems, there are some results concerning the existence of global small smooth solutions, see [1,3,17] and the references cited therein. However, since we consider the half line problem with Dirichlet boundary condition, the methods used for Cauchy problem can not be applied directly in our case.
2. Local existence. The following theorem on the local existence of smooth solutions to (4), (7) and (8) can be established by using the local existence results of initial boundary value problem for hyperbolic systems (cf. [2,7]).
Proof. We shall use the results from [2] to prove the local existence results. We first multiply (4) 1 , (4) 4 by a, c κ respectively and rewrite (4), (7) and (8) as follows where U = U (t, x) = (ω(t, x), v(t, x), θ(t, x), q(t, x)) and and T is an arbitrary but fixed constant. We note that, by Assumption 1.1, A 0 is a positive definite matrix and A 1 is a symmetric matrix. Therefore, system (14) is a first order quasi-linear symmetric hyperbolic system. The local existence theorem for such systems has already been proved, see Theorem 11.2 in [2] for details. We only need to show that the conditions in Theorem 11.2 are satisfied for system (14). For reader's convenience, we list them all and will check the corresponding conditions one by one for our case.
(CH) The matrices are diagonalizable with real eigenvalues of constant multiplicities for (U, ξ) ∈ U × S d−1 .
(NC b ) For all (x, t) ∈ ∂Ω × [0, T ] and for all U ∈ U such that B(U )U = b(x, t), the matrix A(U, ν(x)) is non-singular (where ν(x) denotes the outward unit normal to ∂Ω at point x).
(N b ) The boundary matrix B(U ) is of constant, maximal rank for all (x, t) ∈ ∂Ω × [0, T ] and U ∈ U such that B(U )U = b(x, t) and R n = kerB(U ) ⊕ E s (A(U, ν(x))), with ν an outward normal vector to ∂Ω and E s (A(U, ν)) is the stable subspace of the matrix A(U, ν).
where E − (U, x, ξ, z) is the stable subspace of A(U, x, ξ, z) := A(U, ν(x)) −1 (zI n + iA(U, ξ)), and ν(x) denotes the outward unit normal to ∂Ω at point x; and the same is true for Rez = 0 once the subspace E − has been extended by continuity. In our case, the domain Ω equals to (0, +∞) and the corresponding outward normal vector ν(x)| x=0 = −1. The phase space U = R 4 and d = 1; the boundary matrix B(U ) equals to M and b = 0. We say W is a stable subspace of a matrix A if W is generated by eigenvectors associated with negative eigenvalues of A. Clearly, the condition (CH) is satisfied automatically for symmetric hyperbolic systems. All we need is to check that (NC b ),(N b ) and (UKL b ) are satisfied. Since we get that det A(U, ν) = a 2 c 2 = 0 and thus condition (NC b ) holds. Next, let's check the condition ν)). After some simple calculation, we know that kerM = span{(1, 0, 0, 0), (0, 0, 0, 1)}.
In one dimensional case, T * ∂Ω is trivial. So A(U, ξ) in condition UKL b is zero. In this case, E − (U, x, ξ, z) is the stable subspace of zA(U, ν(x)) −1 . We know from matrix theory that if λ is a eigenvalue of matrix A, then 1 λ is a eigenvalue of matrix A −1 . So, the eigenspace generated by eigenvectors with negative eigenvalues of matrix A is exactly the same with eigenspace generated by eigenvectors with negative eigenvalues of matrix A −1 . This means that the space E − (U, x, ξ, z) equals to the space E s (A(U, ν)) when Rez > 0. Since R 4 = kerM ⊕ E s (A(U, ν)) , M restricted to E s is isomorphism, which means that there exists C such that Thus the condition (UKL b ) is also satisfied. So far ,we have proved all conditions needed in theorem 11.2. So, we can now apply theorem 11.2 to our systems and this complete our proof.
3. Proof of the global existence. Let We first show the following a priori estimates.
where Γ > 0 is a constant independent of T and E 0 .
By using (6), (19), (23), the definition of E(t) and recalling the boundary condition θ| ∂Ω = 0, we can easily check that Now, we intend to estimate the term I 4 . Integrating by part with respect to x, using the same technique before, we get In term of Sobolev imbedding theorem W 1,1 (Ω) → L ∞ , we have Combining the above calculations, we have Next we employ the technique of [24], also see [14] and [21], in dealing with the boundary term on the righthand side of (26). Note that, the methods used in [14] can not be used here since the estimates for θ xt are not expected. However, we shall use the techniques developed for the second sound case, see [21]. Differentiating (4) 2 with respect to t, multiplying v x and integrating, we have (27) Using the boundary condition v t | ∂Ω = 0 and the fact that v tt v 1680 YUXI HU AND NA WANG Next, multiplying (4) 3 by θ xt and integrating with respect to (t, x), we obtain Therefore, combining (28) and (29), we get This implies, with the help of (26), that Now, we intend to estimate the term t 0 v x 2 ds. Multiply (4) 3 by 1 b(ω,θ) v x , integrating with respect to (t, x), and using again (6), (19) and (23), we get The term t 0 Ωã b θv tx dxdt can be estimated as follows. Differentiating (4) 2 with respect to x, multiply byã b θ, integrating, using (19) and (23), we get