Global existence, asymptotic behavior and blow-up of solutions for coupled Klein–Gordon equations with damping terms

Abstract

This paper studies the Cauchy problem for the coupled system of nonlinear Klein–Gordon equations with damping terms. We first state the existence of standing wave with ground state, based on which we prove a sharp criteria for global existence and blow-up of solutions when E ( 0 ) < d . We then introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions for 0 < E ( 0 ) < d and E ( 0 ) ≤ 0 , respectively. Furthermore, we prove the global existence and asymptotic behavior of solutions for the case of potential well family with 0 < E ( 0 ) < d . Finally, a blow-up result for solutions with arbitrarily positive initial energy is obtained.