Well-posedness and exponential stability in binary mixtures theory for thermoviscoelastic diffusion materials

A nonlinear theory is derived for a thermoviscoelastic diffusion composite which is modeled as a binary mixture consisting of two Kelvin–Voigt viscoelastic materials. First, the basic equations of the nonlinear theory of heat and diffusion conducting viscoelastic mixtures are derived in Lagrangian description. Then, the theory is linearized and field equations are given for both anisotropic and isotropic centrosymmetric materials. We establish the necessary and sufficient conditions to get a dissipation inequality for isotropic centrosymmetric materials. With the help of the semigroup theory of linear operators, we establish the suitable framework where the linear problem is well posed. Finally, we prove that generically the solution decreases exponentially thanks to the viscous dissipation effects of the constituents.