LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in H-1 x L-2. In particular, any global in positive times solution is bounded in positive times. The result applies to standard energy subcritical focusing nonlinearities broken vertical bar u broken vertical bar(p-1)u,1< p < (d + 2)/(d - 2) as well as to any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems).