Convergence of global solutions for some classes of nonlinear damped wave equations

TL;DR

This paper investigates the long-term behavior of solutions to nonlinear damped wave equations with time-dependent damping, proving convergence to equilibrium using a Lojasiewicz-Simon inequality.

Contribution

It establishes the convergence of global solutions to equilibrium states for certain nonlinear damped wave equations with time-dependent damping.

Findings

Global solutions converge to equilibrium as time approaches infinity.

The convergence is proved using a Lojasiewicz-Simon type inequality.

Results apply to wave equations with analytic nonlinearity and time-dependent damping.

Abstract

We consider the asymptotic behavior of the soltion to the wave equation with time-dependent damping and analytic nonlinearity. Our main goal is to prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Lojasiewicz-Simon type inequality.