Thermo-elastic waves in a model with nonlinear adhesion

In the context of thermo-elasticity we consider initial boundary value problems governed by parabolic and hyperbolic heat propagations. In particular, we describe the evolution of the temperature and displacement fields in a one dimensional string attached to a rigid substrate through an adhesive layer. This adhesive interaction is characterized by a nonlinear term describing the adhesion force exhibiting discontinuities when a critical value of the displacement is reached, in the limit of parabolic heat propagation. We study the well-posedness of the problem under Neumann boundary conditions in the two different regimes of heat propagation and investigate the long time dynamics.