Global existence and uniform decay for the one-dimensional model of thermodiffusion with second sound
© Zhang and Qin; licensee Springer. 2013
Received: 19 June 2013
Accepted: 4 September 2013
Published: 7 November 2013
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.
Keywordsthermodiffusion second sound global existence exponential decay
where u, , and are the displacement, temperature, and heat flux, , and are the chemical potentials and the associated flux. The boundary conditions (1.7) model a rigidly clamped medium with temperature and chemical potentials held constant on the boundary.
There are many results about the classical thermodiffusion equations. By the method of integral transformations and integral equations, Nowacki , Podstrigach  and Fichera  investigated the initial boundary value problem for the linear homogeneous system. Gawinecki  proved the existence, uniqueness and regularity of solutions to an initial boundary value problem for the linear system of thermodiffusion in a solid body. Szymaniec  proved the - time decay estimates along the conjugate line for the solutions of the linear thermodiffusion system. Using the results from , Szymaniec  obtained the global existence and uniqueness of small data solutions to the Cauchy problem of nonlinear thermodiffusion equations in a solid body. Using the semigroup approach and the multiplier method, Qin et al.  obtained the global existence and exponential stability of solutions for homogeneous, nonhomogeneous and semilinear thermodiffusion equations subject to various boundary conditions. Liu and Reissig  studied the Cauchy problem for one-dimensional models of thermodiffusion and explained qualitative properties of solutions and showed which part of the model has a dominant influence on wellposedness, propagation of singularities, - decay estimates on the conjugate line and the diffusion phenomenon.
If we neglect the diffusion in (1.9), then we obtain the classical thermo-elasticity equations. Today models of type I (classical model of thermo-elasticity), of type II (thermal wave), of type III (visco-elastic damping) or second sound present some classification of models of thermo-elasticity (see, e.g., [3, 10, 11]). By considerations of the total energy equation and comparisons with the models of classical thermo-elasticity and thermodiffusion, we shall propose the linear one-dimensional model of thermodiffusion with second sound as mentioned above. Due to our knowledge, there exist no results for thermodiffusion models with second sound.
Our paper is organized as follows. In Section 2, we present some notations and the main result. Section 3 is devoted to the proof of the main result.
2 Notations and main result
Our main result reads as follows.
Remark 2.1 If the initial value , , ( will be defined later), then the solution , and problem (1.1)-(1.7) yields higher regularity in t.
3 Proof of the main result
We shall divide the proof into two steps: in Step 1, we shall use the semigroup approach to prove the existence of global solutions and the Remark 2.1; Step 2 is devoted to proving the uniform decay of the energy by the boundary control and the multiplier method.
Step 1. Existence of global solutions.
Then by Theorem 2.3.1 of  about the existence and regularities of solutions, we can complete the proof.
Step 2. 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 [10, 11]). We mainly refer to Racke  for the approaches of thermo-elastic models with second sound.
Hence, it follows from (3.36), . Applying (3.34) again, we can conclude (2.5) with . The proof is complete.
This paper was in part supported by the NNSF of China with contract numbers 11031003, 11271066 and a grant from Shanghai Education Commission 13ZZ048.
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