Wave Equations in Modulation Spaces–Decay Versus Loss of Regularity

Abstract

Recently the study of partial differential equations in modulation spaces gained some relevance. A well-known Cauchy problem is that for the wave equation, where several contributions exist concerning the local (in time) well-posedness. We refer to Bényi et al. (J Func Anal 246.2:366–384, 2007), Cordero and Nicola (J Math Anal Appl 353.2:583–591, 2009) and Reich (Modulation spaces and nonlinear partial differential equations. PhD thesis, TU Bergakademie Freiberg, 2017). By taking advantage of some tools and concepts from the theory of partial differential equations the authors provide some time-dependent estimates of the solution u = u(t, x) to the Cauchy problem of the free wave equation. The main result yields the possibility to consider more delicate problems concerning the wave equation in modulation spaces such as global (in time) well-posedness results.