Long time behaviors for 3D cubic damped Klein-Gordon equations in inhomogeneous mediums

TL;DR

This paper investigates the long-term behavior of solutions to 3D damped Klein-Gordon equations in inhomogeneous media, showing convergence to equilibrium states under various conditions using concentration-compact attractors.

Contribution

It establishes the asymptotic dynamics and convergence properties of solutions to damped Klein-Gordon equations in inhomogeneous environments, introducing the concept of concentration-compact attractors.

Findings

Solutions in the defocusing case converge to equilibrium.

In the focusing case, solutions tend to a superposition of equilibria.

Existence of concentration-compact attractors is proven.

Abstract

In this paper, we study the asymptotic dynamics of global solutions to damped Klein-Gordon equations in inhomogeneous mediums (KGI). In the defocusing case, we prove for any initial data, the solution is globally define in forward time and it will converge to an equilibrium. In the focusing case, for global solutions, we prove the solutions converge to the superposition of equilibriums among which there exists at most one equilibrium to KGI and the other equilibriums are solutions to stationary nonlinear Klein-Gordon equations. The core ingredients of our proof are the existence of the "concentration-compact attractor" and the gradient system theory.