Energy decay in thermoelastic diffusion theory with second sound and dissipative boundary

Abstract

We study the energy decay of the solutions of a linear homogeneous anisotropic thermoelastic diffusion system with second sound and dissipative boundary of the form

$$\mathbf{T}(x,t)n(x) = -\gamma_0v(x,t) -\int_0^\infty \lambda(s)v^t(x,s) ds. $$

This boundary condition well describes a material for which the domain outside the body consists in a material of viscoelastic type. Models of boundary conditions including a memory term which produces damping were proposed in Fabrizio and Morro (Arch. Ration. Mech. Anal. 136:359–381, 1996) in the context of Maxwell equations and in Propst and Prüss (J. Integral Equ. Appl. 8:99–123, 1996) for sound evolution in a compressible fluid.

The thermal and diffusion disturbances are modeled by Cattaneo-Maxwell law for heat and diffusion equations to remove the physical paradox of infinite propagation speed in the classical theory within Fourier’s law. The system of equations in this case is a coupling of three hyperbolic equations. By introducing a boundary free energy, we prove that, if the kernel λ exponentially decays in time then also the energy exponentially decays. Finally, we generalize the obtained results to the Gurtin-Pipkin model.