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 [DGL22] 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 [GR14]. 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.(a + b 1 ) 2 ≺ a.Note that a is bounded by √ a as a direct consequence of Glaeser's inequality: If a ∈ C 2 (R), a(x) ≥ 0 for all x ∈ R and a L ∞ ≤ M 1 , then |a (x)| 2 ≤ 2M 1 a(x),