Asymptotics of solutions with a compactness property for the nonlinear damped Klein-Gordon equation

TL;DR

This paper investigates the long-term behavior of solutions to the nonlinear damped Klein-Gordon equation, proving convergence to bound states with specific rates under certain conditions, and providing an example of algebraic decay.

Contribution

It establishes that solutions asymptotically approach bound states, characterizes convergence rates for non-degenerate and degenerate states, and offers an explicit example of algebraic decay.

Findings

Solutions tend to bound states asymptotically.

Convergence occurs at exponential or algebraic rates depending on state degeneracy.

An example demonstrates convergence at rate t^{-1} to an excited state.

Abstract

We consider the nonlinear damped Klein-Gordon equation \[ \partial_{tt}u+2\alpha\partial_{t}u-\Delta u+u-|u|^{p-1}u=0 \quad \text{on} \ \ [0,\infty)\times \mathbb{R}^N \] with $\alpha>0$, $2 \le N\le 5$ and energy subcritical exponents $p>2$. We study the behavior of solutions for which it is supposed that only one nonlinear object appears asymptotically for large times, at least for a sequence of times. We first prove that the nonlinear object is necessarily a bound state. Next, we show that when the nonlinear object is a non-degenerate state or a degenerate excited state satisfying a simplicity condition, the convergence holds for all positive times, with an exponential or algebraic rate respectively. Last, we provide an example where the solution converges exactly at the rate $t^{-1}$ to the excited state.