Long time dynamics for weakly damped nonlinear Klein-Gordon equations

TL;DR

This paper investigates the long-term behavior of weakly damped nonlinear Klein-Gordon equations with time-dependent damping that diminishes to zero, establishing soliton resolution for radial solutions under certain decay conditions.

Contribution

It extends previous results by analyzing damping that decreases over time and employs the Lojasiewicz-Simon inequality instead of invariant manifold theory.

Findings

Soliton resolution holds for radial solutions with slow damping decay

The Lojasiewicz-Simon inequality is effective in this context

Results generalize previous fixed damping cases

Abstract

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.