On fractional and classical hyperbolic obstacle-type problems

Abstract

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 $\nu\searrow 0$. We prove that the weak and very weak solutions of those problems in the fractional setting converge as $s\nearrow 1$ to a weak solution and to a very weak solution, respectively, of the correspondent problems in the classical framework.