Inhomogeneous wave equation with $t$-dependent singular coefficients

TL;DR

This paper investigates the inhomogeneous wave equation with time-dependent singular coefficients, establishing conditions for very weak solutions and exploring numerical solutions in simplified models with discontinuous coefficients.

Contribution

It introduces Levi conditions for very weak solutions of wave equations with singular, time-dependent coefficients and examines numerical approaches in toy models.

Findings

Established Levi conditions for very weak solutions.

Numerical analysis of wave equations with discontinuous coefficients.

Demonstrated existence of solutions in toy models with Heaviside and delta functions.

Abstract

This paper is devoted to the study of the inhomogeneous wave equation with singular (less than continuous) time dependent coefficients. Particular attention is given to the role of the lower order terms and suitable Levi conditions are formulated in order to obtain a very weak solution as introduced in \cite{GR:14}. Very weak solutions for this kind of equations are also investigated from a numerical point of view in two toy models: the wave equation with a Heaviside function and a delta distribution, respectively, as coefficient in its principal part.