Spectral stability analysis for special solutions of second order in time PDEs: The higher dimensional case

We develop a general theory to treat the linear stability of certain special solutions of second order in time evolutionary PDE. We apply these results to standing waves of the following problems: the Klein–Gordon equation, for which we consider both ground states and certain excited states, the Klein–Gordon–Zakharov system and the beam equation. We also discuss applications to excited states for the Klein–Gordon model as well as multidimensional traveling waves (not necessarily homoclinic to zero) for general second order equations of this type. In all cases, our abstract results provide a complete characterization of the linear stability of such solutions.