On the wave-like energy estimates of Klein-Gordon type equations with time dependent potential

TL;DR

This paper investigates conditions under which the energy of Klein-Gordon equations with time-dependent potentials behaves like wave energy, introducing a generalized zero mean condition and modified energy conservation for better understanding.

Contribution

It introduces the generalized zero mean condition and generalized modified energy conservation to analyze energy behavior in Klein-Gordon equations with time-dependent potentials.

Findings

Energy asymptotics depend on the integral of the potential, not just its order.

The generalized zero mean condition determines wave-like energy behavior.

A new framework for energy conservation in variable potential scenarios.

Abstract

We consider the conditions for the time dependent potential in which the energy of the Cauchy problem of Klein-Gordon type equation asymptotically behaves like the energy of the wave equation. The conclusion of this paper is that the condition is not always given by the order of the potential itself, but should be given by "generalized zero mean condition", which is represented by the integral of the potential. We also introduce "generalized modified energy conservation" in order to describe the appropriate energy for our problem.