Attractor Continuity for Semilinear Parabolic Equations on Thin Domains with Degenerating Outward Peaks

TL;DR

This paper investigates how the long-term behavior (attractors) of semilinear parabolic equations on thin, cusp-shaped domains changes as the domain degenerates, establishing conditions for continuity and convergence rates.

Contribution

It provides a rigorous analysis of attractor continuity and convergence rates for equations on thin domains with outward peaks, a previously less understood geometric setting.

Findings

Proves attractor continuity as domain degenerates

Establishes convergence rates of attractors

Analyzes effects of cusp geometry on dynamics

Abstract

In this work, we analyze the asymptotic behavior of the attractors associated with a semilinear parabolic equation subject to homogeneous Neumann boundary conditions and defined on a thin domain $R^\varepsilon \subset \mathbb{R}^{1+n}$. We assume that the thin domain exhibits a cusp, known as an outward peak, whose geometry is characterized by a nonnegative function that vanishes at a point on the boundary. Our objective is to rigorously establish the continuity of the attractors as $\varepsilon \to 0$ and to determine their rate of convergence.