Global existence and blow up of the solution for nonlinear Klein–Gordon equation with general power-type nonlinearities at three initial energy levels

Abstract

This paper investigates the local existence, global existence and finite time blow up of the solution to the Cauchy problem for a class of nonlinear Klein–Gordon equation with general power-type nonlinearities. We give some sufficient conditions on the initial data such that the solution exists globally or blows up in finite time with low initial energy and critical energy. Further a finite time blow up result of the solution with high initial energy is proved.