Remarks on the qualitative behavior of the undamped Klein-Gordon equation

Abstract

We consider the undamped Klein-Gordon equation in bounded domains with homogeneous Dirichlet boundary conditions. For any real value of the initial energy, particularly for supercritical values of the energy, we give sufficient conditions to conclude blow up of weak solutions. The success of the analysis is based in a careful analysis of a differential inequality. Our results improve previous ones in the literature. Our result is extended to a general abstract non-dissipative wave equation.