Study On Qualitative Properties Of Dissipative Klein-gordon Equation With Potential Source Term

In this paper,we study the Cauchy problem of the Klein-Gordon equation withδ（x）potential in the complex domain and the Cauchy problem of the Klein-Gordon equation with potential term and strong and weak damping term in the real domain.The second chapter studies the Cauchy problem of the Klein-Gordon equation withδ（x）potential.This chapter establishes an effective variational framework for such an equation with the characteristics of the Klein-Gordon equation and Schr（?）dinger equation in the complex field,the energy and related functionals are defined,proves energy conservation and mass non-conservation,and some lemmas.In this chapter,the global existence of the solution under the subcritical initial energy level is derived via the bounded of the norm,and the results under the subcritical initial energy level are extended to the critical initial energy level through scale transformation.Due to the non-conservation of mass,the existence of the potential term and the term u_tt,this chapter encounters difficulties in proving the finite time blow up of the solution through the concave function method.We explain the difficulties in proving the blow up by calculating the expression of J（?）（t）.The third chapter studies the initial value problem of the nonlinear Klein-Gordon equation with the dissipation term.In this chapter,by introducing the potential well method,the energy and related functionals are defined.Later,the existence and uniqueness of the local solution are proved by using the Galerkin method,and the boundedness of the norm proves the global existence of the solution under the subcritical initial energy level.The multiplier method admits proves the asymptotic behavior of the solution below the subcritical energy level.After that,this chapter extends the conclusion under the subcritical energy level to the critical energy level according to the dissipative property of the equation.By constructing new auxiliary functionals,the method of concave functions is used to prove the finite time blow up at three different initial energy levels.