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\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.