Instability of Standing Wave for the Klein–Gordon–Hartree Equation

The instability property of the standing wave uω(t, x) = eiωtφ(x) for the Klein-Gordon-Hartree equation $$\frac{{\partial ^2 u}} {{\partial t^2 }} - \Delta u + u - \left( {\left| x \right|^{ - \gamma } *\left| u \right|^2 } \right)u = 0, x \in \mathbb{R}^N , 0 < \gamma < \min \left\{ {N,4} \right\}$$ is investigated. For the case N ≥ 3 and \(\omega ^2 < \tfrac{2} {{N + 4 - \gamma }}\), it is shown that the standing wave eiωtφ(X) is strongly unstable by blow-up in finite time.