We present in detailthe resultsonglobal attraction to stationary states for nonlinear Hamiltonian PDEs in infinite space: for 1Dwave equations coupled to one nonlinear oscillator (the "Lamb system"), for a 1D wave equation coupled to several nonlinearoscillators andfor a 1D wave equation coupled to a continuum of nonlinear oscillators, and for a 3Dwave equation and Maxwell’s equationscoupled to a charged relativistic particle with density of charge satisfying the Wiener condition. In particular, the radiation damping in classical electrodynamics is rigorously proved for the first time. The proofs rely on calculation of energy radiation to infinity and use the concept of omega-limit trajectories and the Wiener Tauberian theorem. The last sectionconcerns3D wave equations with concentrated nonlinearities. The key step in the proof is an investigation of a nonlinear integro-differential equation.

In this chapterwe present thefirst results on global attraction tosolitons forthe scalar wave field and the Maxwell field coupled to the charged relativistic particlewith density of charge satisfying the Wiener condition. In particular, the radiation damping in classical electrodynamics is rigorously proved for the first time (in this chapter, for the translation-invariant case). The proofs rely on calculation of energy radiation to infinity and canonical transformation to the comoving frame and use energy bound from below and the Wiener Tauberian theorem.