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

Abstract

In this paper, we discuss a role of the potential term on the $L^2$ estimate of the solution itself for wave equations with/without a damping term. In the case of free waves, it is known ([10]) that the $L^2$-norm of the solution itself generally increases to infinity (as $t \to \infty$) in $1$ and $2$ dimensions. However, in this paper, we report that such a grow-up property can be controlled by adding a potential term with a generous condition. This idea can also be applied to damped wave equations with potential terms, especially in one dimension, where faster energy decay rates are observed than in the case of ordinary damped wave equaions. Applications to heat and plate equations with a potential can also be treated. In this paper, the low dimensional case is a main target.