On the wave equation with space dependent coefficients: singularities and lower order terms

TL;DR

This paper investigates the wave equation with space-dependent discontinuous coefficients, establishing conditions for very weak solutions and analyzing their qualitative behavior through numerical simulations.

Contribution

It extends previous work to space-dependent coefficients, formulates Levi conditions for lower order terms, and provides a numerical analysis of solutions.

Findings

Existence of very weak solutions under Levi conditions

Numerical simulations illustrating solution behavior

Qualitative analysis of solutions with discontinuous coefficients

Abstract

This paper complements the study of the wave equation with discontinuous coefficients initiated in \cite{DGL:22} in the case of time-dependent coefficients. Here we assume that the equation coefficients are depending on space only and we formulate Levi conditions on the lower order terms to guarantee the existence of a very weak solution as defined in \cite{GR:14}. As a toy model we study the wave equation in conservative form with discontinuous velocity and we provide a qualitative analysis of the corresponding very weak solution via numerical methods.