Existence of a generalized polynomial attractor for the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity

TL;DR

This paper proves the existence of a generalized polynomial attractor for a wave equation with complex damping and nonlinearity, using a novel criterion that handles critical cases without requiring compactness.

Contribution

It introduces a new criterion based on contractive functions for polynomial attractors, applicable to wave equations with nonlocal damping, anti-damping, and critical nonlinearities.

Findings

Existence of polynomial attractor for the wave equation established

Attractor's attractive speed estimated

Criterion applicable to critical nonlinear cases

Abstract

In this paper, we first establish a criterion based on contractive function for the existence of polynomial attractors. This criterion only involves some rather weak compactness associated with the repeated limit inferior and requires no compactness, which makes it suitable for critical cases. Then by this abstract theorem, we verify the existence of a polynomial attractor and estimate its attractive speed for the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity.