Global Well-posedness Of Initial Boundary Value Problem For Klein-Gordon Equation With Logarithmic Source Term

In this paper,we study the initial boundary value problem of Klein-Gordon equation with logarithm source term and damping term,i.e where u0?H01(?),u1?L2(?),T ?(0,+?],?=or 1,p=2 or 2<p<1+n/(n-2),if n?3;2<p<+?,if n=1 or 2.?? Rn is a bounded domain with smooth boundary (?)?.For |u|p-2uln |u|,when p=2,we call it logarithm source term;when p>2,we call it high-order logarithmic source term.We use the Faedo-Galerkin method and the logarithmic Sobolev inequality to obtain the global existence of weak solutions when we deal with logarithmic source term with damping term.Using the convexity method and potential well theory,we study the behavior of the solution at infinity and obtain the energy decay estimate of the solution.Without damping,we use the Faedo-Galerkin method and energy method to obtain the existence and uniqueness of the local solution when we deal with the high order logarithmic source term.Using the convexity method and potential well theory,we study the global existence and the blow-up of the solution in finite time.