Singular Klein-Gordon equation on a bounded domain

TL;DR

This paper studies the wave equation with singular potentials and boundary conditions, establishing existence, uniqueness, and weak solutions using trace and extension domain techniques.

Contribution

It introduces a framework for solving wave equations with singularities in potential, initial data, and boundary conditions, ensuring well-posedness.

Findings

Established existence and uniqueness of weak solutions

Developed methods for handling singularities in wave equations

Proved consistency of very weak solutions

Abstract

In this paper, we consider the wave equation for the Laplace operator with potential, initial data, and nonhomogeneous Dirichlet boundary condition. We establish a weak solution by using traces and extension domains. We also establish the existence, uniqueness and consistency of the very weak solution for the wave equation with singularities in the potential, initial data, source term, boundary and boundary condition.