Existence and uniqueness of the weak solution for a contact problem

We study a mathematical model which describes the antiplane shear deformation of a cylinder in frictionless contact with a rigid foundation. The material is assumed to be electroviscoelastic with longterm memory, the friction is modeled with Tresca s law and the foundation is assumed to be electrically conductive. First we derive the classical variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field, a timedependent variational equation for the potential field and a differential equation for the bounding field. Then we prove the existence of a unique weak solution for the model. The proof is based on arguments of evolution equations and the Banach fixed point theorem.