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 H1×L2. In
particular, any global solution is bounded. The result applies to standard
energy subcritical focusing nonlinearities ∣u∣p−1u,
1\textless{}p\textless{}(d+2)/(d-2) as well as 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)