On the Cauchy problem for the Klein-Gordon equation with the Hartree type semilinear term in the de Sitter spacetime. (Q1984583)

Let \(f(z)=\lambda|z|^p,\lambda|z|^{p-1}z\) and \(V=\lambda|z|^{p+1}/(p+1)\) with \(p>1\), the authors studied the Cauchy problem for the Klein-Gordon equation with the Hartree type semilinear term \(e^{-nHpt}(W*V(u))f(u)\), with \(|W|\le|x|^{-\gamma}\), \(0<\gamma<n\), in the \(1+n\) dimensional de Sitter spacetime with the Hubble constant \(H\in\mathbb{R}\). The effects of the spatial expansion (\(H>0\)) and contraction on the existence of the solution of the equation are studied. In particular, when \(H>0\), under the condition of small data, global existence for \(t\in[0,\infty)\) as well as the asymptotic behavior as \(t\to +\infty\) is shown.