From Wave to Klein—Gordon Type Decay Rates

Abstract

The authors consider Klein-Gordon equations in $\mathbb{R}^n$ with time-dependent coefficients. Their aim is to provide $L^p$---$L^q$ decay estimates. Contrary to the classical case, in addition to Klein-Gordon type decay rates, wave type decay rates may appear, or it is even possible not to have $L^p$---$L^q$ decay at all. In the latter case, the solutions have a Floquet behavior, meaning that the energy cannot be controlled by time-dependent functions with a suitable growth as time tends to infinity.