In this paper we consider the hyperbolic-parabolic system of equations describing the thermodiffusion in a solid body. The theory of thermodiffusion is a combination of the theory of heat conduction and diffusion with the theory of elasticity. In this work, we establish the well-posedness of an intial-boundary value problem for a system describing this phenomenon, using the semigroup theory. Moreover, we prove a polynomial decay of solutions satisfying certain properties.

In this paper, we consider a coupled system of two biharmonic equations with damping and source terms of variable-exponents nonlinearities, supplemented with initial and mixed boundary conditions. We establish an existence and uniqueness result of a weak solution, under suitable assumptions on the variable exponents. Then, we show that solutions with negative-initial energy blow up in finite time. To illustrate our theoritical findings, we present two numerical examples.