Global existence, blow-up and asymptotic behavior of solutions for a class of coupled nonlinear Klein-Gordon equations with damping terms
The author investigates the global existence, asymptotic behavior and blow-up of the solutions of the initial value problem to the coupled equations \[ \begin{cases} u_{tt}-\Delta u+m_1^2u+\gamma _1u_t=(| u| ^{2p}+| v| ^{p+1}| u| ^{p-1})u,\\ v_{tt}-\Delta v+m_2^2v+\gamma _2v_t=(| v| ^{2p}+| u| ^{p+1}| v| ^{p-1})v \end{cases} \] for \(t>0\), \(x\in \mathbb{R}^n\), where \(n\geq 2\), \(m_1,m_2,\gamma _1,\gamma _2>0\), \(p>0\) for \(n=2\) and \(0<p<2/(n-2)\) for \(n\geq 3\). He introduces the family of potential wells, and with use of that he presents a sharp criterion for global existence or blow-up of the solutions.
scientific article; zbMATH DE number 6118576
scientific article