Exponential decay for a thermo-viscoelastic dynamic contact problem with friction and infinite memory

We investigate a dynamic thermo-viscoelastic contact problem involving a deformable body in frictional interaction with a thermally conductive foundation. The model incorporates infinite memory effects through a hereditary integral term, a thermal normal damped response law, Tresca's friction condition, and a regularized heat exchange mechanism at the contact interface. We first derive a weak variational formulation of the problem and establish the existence and uniqueness of weak solutions. Global solvability is then obtained using nonlinear semigroup theory. We prove the exponential stability of the associated energy, showing that solutions converge exponentially to equilibrium. Finally, these analytical results are illustrated through simplified numerical simulations inspired by the thermo-viscoelastic dynamic contact problem with infinite memory under consideration.