Long time dynamics for weakly damped nonlinear Klein-Gordon equations

We continue our study of damped nonlinear Klein-Gordon equations. In our previous work we considered fixed positive damping and proved a form of the soliton resolution conjecture for radial solutions. In contrast, here we consider damping which decreases in time to 0. In the class of radial data we again establish soliton resolution provided the damping goes to 0 sufficiently slowly. While our previous work relied on invariant manifold theory, here we use the Lojasiewicz-Simon inequality applied to a suitable Lyapunov functional.