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

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 Pruss (J. Integral Equ. Appl. 8:99–123, 1996) for sound evolution in a compressible fluid.