Some attempts on $L^{2}$ boundedness for 1-D wave equations with time variable coeffecients

TL;DR

This paper investigates the $L^2$-boundedness of solutions to one-dimensional wave equations with time-dependent speeds, using a multiplier method tailored to the unique properties of 1D space.

Contribution

It introduces a simple multiplier approach leveraging 1D space properties to analyze $L^2$ boundedness for wave equations with variable coefficients.

Findings

Established $L^2$ boundedness under certain conditions.

Provided a new method applicable to 1D wave equations with time-dependent coefficients.

Demonstrated the effectiveness of the multiplier technique in this context.

Abstract

We consider the $L^2$-boundedness of the solution itself of the Cauchy problem for wave equations with time-dependent wave speeds. We treat it in the one-dimensional Euclidean space. To study these, we adopt a simple multiplier method by using a special property equiped with the one dimensional space.