GLOBAL EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A KLEIN-GORDON EQUATION WITH EXPONENTIAL TYPE NONLINEAR TERM

GLOBAL EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A KLEIN-GORDON EQUATION WITH EXPONENTIAL TYPE NONLINEAR TERM

H. Ta┼čkesen

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Abstract

In this paper, the global existence and nonexistence of solutions for a Klein- Gordon equation, appearing in a variety of physical situations, with exponential type source term and supercritical initial energy (E(0) > d) are investigated in a bounded domain . In the framework of potential well, a functional including both of initial data is de ned and by sign invariance of this functional the global existence of weak solutions in the case of high initial energy is proved. Moreover, under some conditions imposed on initial displacement and initial velocity a nite time blow up result is provided which extends a result given in the literature.

Keywords

Klein-Gordon equation; Exponential nonlinearity; High initial energy level; Global solution; Blow-up.