A role of potential on L^{2}-estimates for some evolution equations

TL;DR

This paper investigates how potentials influence L^{2}-estimates for wave equations, showing that potentials can control solution growth and enable decay in low-dimensional cases, with implications for heat and plate equations.

Contribution

It demonstrates that adding potentials under certain conditions can control solution growth and induce decay in low-dimensional wave equations, extending to heat and plate equations.

Findings

Potentials can control L^{2}-norm growth in wave equations.

Adding potentials enables decay estimates in low dimensions.

Applications to heat and plate equations are discussed.

Abstract

In this papwe we consider an effective role of the potential of the wave equations with/without damping on the L^{2}-estimate of the solution itself. In the free wave equation case it is known that the L^{2}-norm of the solution itself generally grows to infinity (as time goes to infinity) in the one and two dimensional cases, however, by adding the potential with quite generous conditions one can controle the growth property to get the L^{2}-bounds. This idea can be also applied to the damped wave equations with potential in order to get fast energy and L^{2} decay results in the low dimensional case, which are open for a long period. Applications to heat and plate equations with a potential can be also studied. In this paper the low dimensional case is a main target.