Strong attractors for weakly damped quintic wave equation in bounded domains

TL;DR

This paper investigates the long-term behavior of a weakly damped wave equation with quintic non-linearity in bounded domains, establishing the existence of a strong global attractor with finite fractal dimension.

Contribution

It proves the existence of a strong global attractor with finite fractal dimension for the weakly damped quintic wave equation in bounded domains, using Strichartz estimates and quasi-stable estimation.

Findings

Existence of a strong global attractor in the phase space.

Finite fractal dimension of the attractor established.

Utilization of Strichartz estimates for bounded domains.

Abstract

In this paper, we study the longtime dynamics for the weakly damped wave equation with quintic non-linearity in a bounded smooth domain of $\mathbb{R}^3.$ Based on the Strichartz estimates for the case of bounded domains, we establish the existence of a strong global attractor in the phase space $H^2(\Omega)\cap H^1_0(\Omega)\times H^1_0(\Omega)$. Moreover, the finite fractal dimension of the attractor is also shown with the help of the quasi-stable estimation.