On fractional and classical hyperbolic obstacle-type problems

TL;DR

This paper investigates the behavior of solutions to obstacle-type wave problems in fractional and classical frameworks, demonstrating convergence of solutions as fractional order approaches 1 and viscosity vanishes.

Contribution

It introduces a fractional framework for obstacle wave problems and establishes convergence results linking fractional and classical solutions as parameters vary.

Findings

Weak solutions for viscous problems approximate very weak solutions for inviscid problems.

Solutions in the fractional setting converge to classical solutions as fractional order approaches 1.

The approach bridges fractional and classical obstacle wave 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.