On the Wave Equation with Space Dependent Coefficients: Singularities and Lower Order Terms

Abstract

This paper complements the study of the wave equation with discontinuous coefficients initiated in (Discacciati et al. in J. Differ. Equ.319 (2022) 131–185) 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 (Garetto and Ruzhansky in Arch. Ration. Mech. Anal.217 (2015) 113–154). 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.