On fractional and classical hyperbolic obstacle-type problems

We consider weak solutions for the obstacle-type viscoelastic (nu > 0) and very weak solutions for the obstacle inviscid (nu = 0) Dirichlet problems for the heterogeneous and anisotropic wave equation in a fractional framework based on the Riesz fractional gradient D-s (0 < s < 1). We use weak solutions of the viscous problem to obtain very weak solutions of the inviscid problem when nu SE arrow 0. We prove that the weak and very weak solutions of those problems in the fractional setting converge as s NE arrow 1 to a weak solution and to a very weak solution, respectively, of the correspondent problems in the classical framework.