Asymptotic decomposition for nonlinear damped Klein-Gordon equations

TL;DR

This paper demonstrates that solutions to nonlinear damped Klein-Gordon equations decompose into a finite set of equilibrium points over time, using concentration-compact attractors and damping effects.

Contribution

It introduces the concept of a concentration-compact attractor for damped Klein-Gordon equations and proves solutions asymptotically approach equilibrium points.

Findings

Solutions decompose into finite equilibrium points

Existence of concentration-compact attractor established

Profiles are shown to be equilibrium points

Abstract

In this paper, we proved that if the solution to damped focusing Klein-Gordon equations is global forward in time, then it will decouple into a finite number of equilibrium points with different shifts from the origin. The core ingredient of our proof is the existence of the "concentration-compact attractor" which yields a finite number of profiles. Using damping effect, we can prove all the profiles are equilibrium points.