# Global attraction to solitary waves for Klein–Gordon equation with mean field interaction

### Alexander Komech

Faculty of Mathematics, University of Vienna, Wien A-1090, Austria, Institute for Information Transmission Problems, Moscow 101447, Russia### Andrew Komech

Institute for Information Transmission Problems, Moscow 101447, Russia, Mathematics Department, Texas A&M University, College Station, TX, USA

## Abstract

We consider a $\mathbf{U}(1)$-invariant nonlinear Klein–Gordon equation in dimension $n⩾1$, self-interacting via the mean field mechanism. We analyze the long-time asymptotics of finite energy solutions and prove that, under certain generic assumptions, each solution converges as $t\rightarrow \pm \infty$ to the two-dimensional set of all “nonlinear eigenfunctions” of the form $\phi (x)e^{−i\omega t}$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.

## Cite this article

Alexander Komech, Andrew Komech, Global attraction to solitary waves for Klein–Gordon equation with mean field interaction. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 3, pp. 855–868

DOI 10.1016/J.ANIHPC.2008.03.005