Global existence and non-existence analyses to a nonlinear Klein-Gordon system with damping terms under positive initial energy

Abstract

This paper considers the Cauchy problem for a nonlinear Klein-Gordon system with damping terms. In the existing works, the solution with low and critical initial energy was studied. We extend the previous results on following three aspects. Firstly, we consider the vacuum isolating phenomenon of solution under initial energy $ E(0)\leq0 $. We find that the corresponding vacuum region is an ball and it expands to whole phase space as $ E(0) $ decays to $ -\infty $. Secondly, we discuss the asymptotic behavior of blow-up solution and prove that the solution grows exponentially. The growth speed is estimated especially. Finally, the solution with arbitrary positive initial energy is studied. In this case, the initial conditions such that the solution exists globally and blows up in finite time are given, respectively.