Convergence of non-autonomous attractors for subquintic weakly damped wave equation

TL;DR

This paper investigates the long-term behavior of solutions to a non-autonomous weakly damped wave equation with subquintic nonlinearity, establishing the existence and convergence of various attractors under certain conditions.

Contribution

It demonstrates the existence, smoothness, and convergence of pullback, uniform, and cocycle attractors for the equation, extending understanding of non-autonomous dynamical systems with subquintic growth.

Findings

Existence of pullback, uniform, and cocycle attractors.

Smoothness properties of the attractors.

Upper-semicontinuous convergence to the autonomous attractor.

Abstract

We study the non-autonomous weakly damped wave equation with subquintic growth condition on the nonlinearity. Our main focus is the class of Shatah--Struwe solutions, which satisfy the Strichartz estimates and are coincide with the class of solutions obtained by the Galerkin method. For this class we show the existence and smoothness of pullback, uniform, and cocycle attractors and the relations between them. We also prove that these non-autonomous attractors converge upper-semicontinuously to the global attractor for the limit autonomous problem if the time-dependent nonlinearity tends to time independent function in an appropriate way.