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

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Global existence and uniform decay for the one-dimensional model of thermodiffusion with second sound Ming Zhang1* and Yuming Qin2 * Correspondence: [email protected] 1 College of Information Science and Technology, Donghua University, Shanghai, 201620, P.R. China Full list of author information is available at the end of the article

Abstract 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

1 Introduction 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: ρutt – (λ + μ)uxx + γ θx + γ θx =  in (, ) × (, +∞), √ cθt + kqx + γ utx + dθt =  in (, ) × (, +∞), √ nθt + Dqx + γ utx + dθt =  in (, ) × (, +∞), √ τ qt + q + kθx =  in (, ) × (, +∞), √ τ qt + q + Dθx =  in (, ) × (, +∞),

(.) (.) (.) (.) (.)

together with the initial conditions ⎧ ⎪ ⎪ ⎨u(x, ) = u (x), θ (x, ) = θ (x), ⎪ ⎪ ⎩ q (x, ) = q (x),

ut (x, ) = u (x), θ (x, ) = θ (x),

(.)

q (x, ) = q (x),

and the boundary conditions u(x, t)|x=, = θ (x, t)|x=, = θ (x, t)|x=, = ,

(.)

where u, θ , and q are the displacement, temperature, and heat flux, θ , and q are the chemical potentials and the associated flux. The boundary conditions (.) model a rigidly ©2013 Zhang and Qin; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Zhang and Qin Boundary Value Problems 2013, 2013:222 http://www.boundaryvalueproblems.com/content/2013/1/222

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clamped medium with temperature and chemical potentials held constant on the boundary. 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 nc – d > .

(.)

The classical thermodiffusion equations were first given by Nowacki [, ] in . The equations describe the process of thermodiffusion in a solid body (see, e.g., [–]): ⎧ ⎪ ⎪ ⎨ρutt – (λ + μ)uxx + γ θx + γ θx = , cθt – kθxx + γ utx + dθt = , ⎪ ⎪ ⎩ nθt – Dθxx + γ utx + dθt = .

(.)

There are many results about the classical thermodiffusion equations. By the method of integral transformations and integral equations, Nowacki [],