Generalized energy conservation for Klein--Gordon type equations

Abstract

The aim of this paper is to derive energy estimates for solutions of the Cauchy problem for the Klein--Gordon type equation $u_{tt} - \bigtriangleup u + m(t)^{2} u = 0$. The coefficient $m$ is given by $m(t)^{2} = \lambda(t)^{2} + p(t)$ with a decreasing, smooth shape function $\lambda$ and an oscillating, smooth and bounded perturbation function $p$. We study under which assumptions for $\lambda$ and $p$ one can expect results about a generalization of energy conservation. The main theorems of this note deal with $m$ belonging to $C^{M}$, $M \ge 2$, and $m$ belonging to the Gevrey class $\gamma^{(s)}$, $s \ge 1$.