On the Cauchy problem for the nonlinear Klein-Gordon equation with a cubic convolution

Abstract

We study the Cauchy problem for the nonlinear Klein-Gordon equation with a cubic convolution $\{V_{\gamma}*(w(t))^{2}\}w(t)$, where $V_{\gamma}(x)=|x|^{-\gamma}$, in $(x,t) \in \mathbf{R}^{n}\times \mathbf{R}$. We prove the existence of weak solutions for $0 \lt \gamma \lt$. We also prove that for $0\lt\gamma\lt{\rm Min}\{4, n\}$ the weak solution is unique and there exists a regular solution.