Nonexistence of global solutions of a class of coupled nonlinear Klein-Gordon equations with nonnegative potentials and arbitrary initial energy

Abstract

In the paper we consider the nonexistence of global solutions of the Cauchy problem for coupled Klein-Gordon equations of the form \begin{eqnarray*} \left\{\begin{array}{l} u_{tt}-Δu+m_1^2 u+K_1(x)u=a_1|v|^{q+1}|u|^{p-1}u v_{tt}-Δv+m_2^2 u+K_2(x)v=a_2|u|^{p+1}|v|^{q-1}v u(0,x)=u_0; u_t(0,x)=u_1(x) v(0,x)=v_0; v_t(0,x)=v_1(x) \end{array} \right. \end{eqnarray*} on $\R\times\R^n$. Firstly for some special cases of $n=2,3$, we prove the existence of ground state of the corresponding Lagrange-Euler equations of the above equations. Then we establish a blow up result with low initial energy, which leads to instability of standing waves of the system above. Moreover as a byproduct we also discuss the global existence. Next based on concavity method we prove the blow up result for the system with non-positive initial energy in the general case: $n\geq 1$. Finally when the initial energy is given arbitrarily positive, we show that if the initial datum satisfies some conditions, the corresponding solution blows up in a finite time.