Global existence and uniform decay for the one-dimensional model of thermodiffusion with second sound

In this paper, we investigate an initial boundary value problem for the one-dimensional linear model of thermodiffusion with second sound in a bounded region. Using the semigroup approach, boundary control and the multiplier method, we obtain the existence of global solutions and the uniform decay estimates for the energy. MSC: 35B40, 35M13, 35Q79.

Here, we denote by λ, μ the material constants, ρ the density, γ  , γ  the coefficients of thermal and diffusion dilatation, k, D the coefficients of thermal conductivity, n, c, d the coefficients of thermodiffusion, and τ  , τ  the (in general very small) relaxation time. All the coefficients above are positive constants and satisfy the condition The classical thermodiffusion equations were first given by Nowacki [, ] in . The equations describe the process of thermodiffusion in a solid body (see, e.g., [-]): Our paper is organized as follows. In Section , we present some notations and the main result. Section  is devoted to the proof of the main result. http://www.boundaryvalueproblems.com/content/2013/1/222

Notations and main result
Let = (, ) and The associated first-order and second-order energy is defined by The energy E(t) is defined by Our main result reads as follows.
Then problem (.)-(.) has a unique global solution such that Moreover, the associated energy E(t) defined by (.) decays exponentially, i.e., there exist positive constants c  and C  such that , and problem (.)-(.) yields higher regularity in t.

Proof of the main result
We shall divide the proof into two steps: in Step , we shall use the semigroup approach to prove the existence of global solutions and the Remark .; Step  is devoted to proving the uniform decay of the energy by the boundary control and the multiplier method.
Step . Existence of global solutions. The proof is based on the semigroup approach (see [, ]) that can be used to reduce problem (.)-(.) to an abstract initial value problem for a first-order evolution equation. In order to choose proper space for (.)-(.), we shall consider the static system associ-http://www.boundaryvalueproblems.com/content/2013/1/222 ated with them (see []). Considering the energy and the property of operator A, we can choose the following state space and the domain of operator A for problem (.)-(.): Using the same method as in [, ], we can prove that the operator A generates a C semigroup of contractions on the Hilbert space H. Define Then by Theorem .. of [] about the existence and regularities of solutions, we can complete the proof.
Step . Uniform decay of the energy.
In this section, we shall assume the existence of solutions in the Sobolev spaces that we need for our computations. The proof of uniform decay is difficult. It is necessary to construct a suitable Lyapunov function and to combine various techniques from energy method, multiplier approaches and boundary control (see [, ]). We mainly refer to Racke [] for the approaches of thermo-elastic models with second sound.