Continuous Dependence for the Linear Differential Equations of Thermo-Diffusion

In this paper, we establish the structural stability for the linear differential equations of thermo-diffusion in a semi-infinite pipe flow. Using the tech-nology of a second-order differential inequality, we prove the continuous dependence on the density ρ and the coefficient of thermal conductivity K. These results show that small changes for these coefficients can’t cause tre-mendous changes for the solutions.


Introduction
The question of continuous dependence of solutions of problems in partial differential equations on coefficients in the equations has been extensively studied in recent years for a variety of problems. This is sometimes referred to as the question of structural stability and numerous references may be found, for instance, in the book of Ames and Straughan [1] and the monograph of Straughan [2]. For more papers one can see [3]- [8]. In structural stability the emphasis is on continuous dependence (convergence result) on changes in the model itself rather than on the initial data. This means changes in coefficients in the partial differential equations and changes in the equations and may be reflected physically by changes in constitutive parameters. What's more, the inevitable error that arises in both numerical computation and the physical measurement of data can exist. It is relevant to know the magnitude of the effect of such errors on the solution.
In the 1970s, W. Nowacki in his papers [9] [10] gave the differential equations of thermodiffusion in one dimensional space and many papers in the literatures have studied this system. For example, [10] [11] [12] investigated the initial-boundary value problem for the linear system of thermodiffusion using different arguments. [13] proved the existence, uniqueness and regularity of the solution to the initial-boundary value problems for the linear system of thermodiffusion in a solid body. L p -L q time decay estimates for the solution of the associated linear Cauchy problem were obtained by [14]. However, in this paper, we considered the differential equations of thermodiffusion in three dimensions and we study not only the continuous dependence on the coefficients of the equations, but also the spatial decay estimates for the solution of the system. In fact, much has been written on the subject of spatial decay bounds for various systems of differential equations, e.g., for a review of such works on Saint-Venant's principle, one can refer to [15]- [23] and the papers cited therein. Recently, there are some new results about structural stability, one could see [24]- [28].
We shall assume that a transient flow occupies the interior of a semi-infinite cylindrical pipe R with boundary ∂R . The pipe has arbitrary cross section denoted by D and the boundary ∂D and the generators of the pipe are parallel to the 3 x axis. We introduce the notations: where z is a running variable along the 3 x axis. Clearly, 0 = R R and 0 = D D.
Let i u , T , and C denote the displacement, temperature, and chemical potential as independent fields, respectively. These fields depend on the space variable ( ) , , x x x and the time variable t and satisfy the following system of equations: with the initial-boundary conditions , , , , , , uniformly in , , as .
In Equations (1.1)-(1.3), ∆ is the Laplacian operator; ρ represents the density; 1 γ and 2 γ are the coefficients of thermal and diffusion dilatation; λ and ν are the material coefficients; K is the coefficient of thermal conductivity; M is the coefficient of diffusion. , , n c d are the coefficients of thermodiffusion.
All the above constants are positive and satisfy satisfies the following estimates ( ) ( ) ( ) where we have supposed that i u  vanish at 0 t = . Similarly, we have where we also have assumed that Following the method used in [31], we can get ( ) Also, we employ the argument used in [31] to get that ( ) 0,t E  may be bounded by known data. Since 2 cn d > , we note again 3) with same initial-boundary conditions, but for different parameters ρ and * ρ , respectively. Define the difference variables as with the initial-boundary conditions We define a new function satisfies the following estimates: Proof. In this section we compare the solutions of the following two problems with the initial-boundary conditions with the same initial-boundary conditions (3.4).
Our goal in this section is to derive the continuous dependence on the param- Similarly, we have η θ θ η γ θ η We make the choice of