Global attractors for the damped nonlinear wave equation in unbounded domains

TL;DR

This paper proves the existence of global attractors for certain damped nonlinear wave equations in unbounded domains using tail estimation, overcoming non-compactness issues in traditional Sobolev spaces.

Contribution

It introduces a method to establish global attractors for wave equations in unbounded domains within standard Hilbert spaces, avoiding restrictive weighted Sobolev space assumptions.

Findings

Global attractors exist for wave equations in unbounded domains.

The tail estimation method effectively handles non-compactness.

Results apply to equations with and without mass terms.

Abstract

The existence of a global attractor for wave equations in unbounded domains is a challenging problem due to the non-compactness of the Sobolev embeddings. To overcome this difficulty, some authors have worked with weighted Sobolev spaces which restrict the choice of the initial data. Using the "tail estimation method" introduced by B. Wang for reaction diffusion equations, we establish in this paper the existence of a global attractor for two wave equations in the traditional Hilbert spaces $\displaystyle H^1(\Omega)\times L^2(\Omega)$ where $\Omega$ is an unbounded domain of $\R^N$. The first equation, with a mass term is studied in the whole space $\R^N$ and the second one without mass term is considered in a domain bounded in only one direction so that Poincar\'e inequality will hold.