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

Abstract

The instability property of the standing wave u ω (t, x) = eiωt φ(x) for the Klein-Gordon-Hartree equation ∂ 2 u ∂t 2 −Δu+u− (|x| −γ *|u| 2 )u = 0, x ∈R N , 0<γ<min{N,4} is investigated. For the case N ≥ 3 and ω 2 <2 N+4−γ, it is shown that the standing wave eiωt φ(X) is strongly unstable by blow-up in finite time.