Wave equation for Sturm-Liouville operator with singular potentials

TL;DR

This paper investigates the wave equation involving Sturm-Liouville operators with singular potentials, establishing existence, uniqueness, and consistency of very weak solutions using spectral asymptotics and separation methods.

Contribution

It introduces a framework for solving wave equations with irregular potentials, extending classical results to singular coefficient cases.

Findings

Proved existence and uniqueness of very weak solutions.

Developed asymptotic analysis for eigenvalues and eigenfunctions.

Established conditions for solution consistency.

Abstract

The paper is denoted to the initial-boundary value problem for the wave equation with the Sturm-Liouville operator with irregular (distributive) potentials. To obtain a solution to the equation, the separation method and asymptotics of the eigenvalues and eigenfunctions of the Sturm-Liouville operator are used. Homogeneous and inhomogeneous cases of the equation are considered. Next, existence, uniqueness, and consistency theorems for a very weak solution of the wave equation with singular coefficients are proved.