Long-time dynamics of the wave equation with nonlocal weak damping and sup-cubic nonlinearity in 3-D domains

TL;DR

This paper investigates the long-term behavior of solutions to a 3-D wave equation with nonlocal weak damping and super-cubic nonlinearity, establishing global well-posedness and the existence of attractors.

Contribution

It proves global well-posedness and attractor existence for the wave equation with nonlocal damping and super-cubic nonlinearity in bounded domains.

Findings

Global well-posedness of Shatah-Struwe solutions

Existence of a global attractor

Existence of a polynomial attractor

Abstract

In this paper, we study the long-time dynamics for the wave equation with nonlocal weak damping and sup-cubic nonlinearity in a bounded smooth domain of $\mathbb{R}^3.$ Based on the Strichartz estimates for the case of bounded domains, we first prove the global well-posedness of the Shatah-Struwe solutions. Then we establish the existence of the global attractor for the Shatah-Struwe solution semigroup by the method of contractive function. Finally, we verify the existence of a polynomial attractor for this semigroup.