On the Cauchy problem for the nonlinear Klein-Gordon equation with a cubic convolution (Q1121453)

We study the Cauchy problem for the nonlinear Klein-Gordon equation with a cubic convolution: \[ \partial^ 2_ tw(t)-\Delta w(t)+w(t)+\{V_{\gamma}*f(w(t))\}w(t)=0;\quad w(0)=\phi (x),\quad \partial_ tw(0)=\psi (x), \] where \(f(w)=w^ 2\), \(V_{\gamma}(x)=| x|^{-\gamma}\) in \((x,t)\in {\mathbb{R}}^ n\times {\mathbb{R}}.\) We prove the existence of weak solutions for \(0<\gamma <n\). We also prove that for \(0<\gamma <Min\{4,n\}\) the weak solution is unique and there exists a regular solution.