Nonlinear Waves in Adhesive Strings

We study a one-dimensional semilinear wave equation modeling the dynamic of an elastic string interacting with a rigid substrate through an adhesive layer. The constitutive law of the adhesive material is that it is assumed to be elastic up to a finite critical state, and beyond such a value the stress discontinuously drops to zero. Therefore, the semilinear equation is characterized by a source term presenting jump discontinuity. Well-posedness of the initial boundary value problem of Neumann type is investigated.