Global attractors for the one dimensional wave equation with displacement dependent damping

TL;DR

This paper investigates the long-term dynamics of a one-dimensional wave equation with nonlinear damping, establishing the existence of a global attractor when damping is positive near zero.

Contribution

It proves the existence of a global attractor for the wave equation with nonlinear damping under specific positivity conditions.

Findings

Existence of a global attractor for the wave equation with nonlinear damping.

The attractor exists if the damping coefficient is positive near zero.

Provides conditions under which long-term behavior is predictable.

Abstract

We study the long-time behavior of solutions of the one dimensional wave equation with nonlinear damping coefficient. We prove that if the damping coefficient function is strictly positive near the origin then this equation possesses a global attractor.