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.
Keywords:thermodiffusion; second sound; global existence; exponential decay
In this paper, we investigate the global existence and uniform decay rate of the energy for solutions for the one-dimensional model of thermodiffusion with second sound:
together with the initial conditions
and the boundary conditions
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.
Here, we denote by λ, μ the material constants, ρ the density, , the coefficients of thermal and diffusion dilatation, k, D the coefficients of thermal conductivity, n, c, d the coefficients of thermodiffusion, and , the (in general very small) relaxation time. All the coefficients above are positive constants and satisfy the condition
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
The associated first-order and second-order energy is defined by
The energy is defined by
Our main result reads as follows.
Theorem 2.1Assume that , , , . Then problem (1.1)-(1.7) has a unique global solution such that
Moreover, the associated energy defined by (2.4) decays exponentially, i.e., there exist positive constants and such that
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.
The proof is based on the semigroup approach (see [4,12]) that can be used to reduce problem (1.1)-(1.7) to an abstract initial value problem for a first-order evolution equation. In order to choose proper space for (1.1)-(1.7), we shall consider the static system associated 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 (1.1)-(1.7):
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.
Multiplying (1.1) by , (1.2) by , (1.3) by , (1.4) by , and (1.5) by in , respectively, and summing up the results, yields
Similarly, we can get
Multiplying (1.1) by in , we get
Multiplying (1.2) by , (1.3) by in and summing them up, yields
Combining (3.5) with (3.6), we get
Now, we conclude from (1.1), (1.4), (1.5) and Poincaré inequality
Multiplying (1.1) by u in , we obtain
From (1.2) and (1.3), we get
Multiplying (3.10) by , (3.11) by in , and summing up the results, we get
The boundary terms are estimated as follows.
for some ,
Combining (3.13)-(3.16), we get
Differentiating (1.1) with respect to t and multiplying by , where
Multiplications of (1.2) by and (1.3) by yield
which implies, using (1.4) and (1.5),
Combining (3.19)-(3.21), we conclude
Using (1.2) and (1.3), we get
With (3.17) and (3.23), we can estimate
Define . Multiplying both sides of (3.12) by ξ and combining the result with (3.7) and (3.24), we obtain for sufficiently small the estimate
where , and . Now, we can define the desired Lyapunov functional . For , to be determined later on, let
Then we conclude from (3.3), (3.4), and (3.15)
By using (3.8), we arrive at
while (3.9) yields
Combining (3.27)-(3.29), we conclude
We choose such that all terms on the right-hand side of (3.30) become negative,
Choosing ε as in (3.31), we obtain from (3.30)
There exist positive constants , and such that for any and , it holds
where , are determined later on. In fact,
At this point, we choose , . Finally, we choose . Thus, we have the validity of (3.34). Combining (3.33) with (3.34), we get
Hence, it follows from (3.36), . Applying (3.34) again, we can conclude (2.5) with . The proof is complete.
The authors declare that they have no competing interests.
The paper is a joint work of all authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
This paper was in part supported by the NNSF of China with contract numbers 11031003, 11271066 and a grant from Shanghai Education Commission 13ZZ048.
(in press)Publisher Full Text
Qin, Y, Zhang, M, Feng, B, Li, H: Global existence and asymptotic behavior of solutions for thermodiffusion equations. J. Math. Anal. Appl.. 408, 140–153 (2013). Publisher Full Text
Gawinecki, JA, Sierpinski, K: Existence, uniqueness and regularity of the solution of the first boundary initial value problem for the equation of the thermodiffusion in a solid body. Bull. Pol. Acad. Sci., Tech. Sci.. 30, 541–547 (1982)
Racke, R: Thermoelasticity with second sound-exponential stability in linear and non-linear 1-d. Math. Methods Appl. Sci.. 25, 409–441 (2002). Publisher Full Text