Cauchy Problem Of Nonlinear KG Equation With Damping Term

Klein-Gordon equation is one of famous evolution equations arising in relativistic quantum mechanics. It describes theÏ€partical field equation. In this paper, we study the Cauchy problem for nonlinear Klein-Gordon equation with damping term1. First we introduce a potential well, we can get the properties of potential well in lemma form. Then we generalize the family of potential wells, and by the definition of the family of potential wells we prove their properties. So we obtain the differences of the single potential well and a family of pote- ntial wells.2. We discuss the invariance of some sets of solutions under the flow of problem, so we prove that the sets of solutions are invariant in the existence time. In addition, we obtain vacuum isolating behavior of solutions for problem,i.e. there exists a vacuum region such that there are no solutions of problem. The vacuum region becomes bigger and bigger with the decreasing of e.3. We prove the global existence and finite time blow up of solutions for problem by using the family of potential wells. At the same time we can give a threshold result for the global existence and nonexistence of solutions.4. We prove the global existence of solutions for problem with critical initial condition I(u₀)≥0, E (0) = d by using Galerkin method and a potential well theory.5. We discuss the asymptotic behavior of the solutions for problem by using energy estimate method and a family of potential wells.