On global attraction to solitary waves. Klein–Gordon equation with mean field interaction at several points

We consider the U(1)-invariant Klein–Gordon equation in dimension n⩾3, self-interacting via the mean field mechanism in finitely many regions. We prove that, under certain generic assumptions, each solution converges as t→±∞ to the two-dimensional set of all “nonlinear eigenfunctions” of the form ϕ(x)e−iωt. The proof is based on the analysis of omega-limit trajectories. The Titchmarsh Convolution Theorem allows us to prove that the time spectrum of any omega-limit trajectory of each finite energy solution consists of a single point. This proves the convergence to the attractor in local sub-energy norms.