Pullback and uniform attractors for nonautonomous reaction-diffusion equation in Dumbbell domains

TL;DR

This paper investigates the long-term behavior of nonautonomous reaction-diffusion equations in Dumbbell-shaped domains, establishing existence of attractors and bounds as the domain's channel collapses.

Contribution

It introduces a novel analysis of reaction-diffusion equations in Dumbbell domains, proving the existence of pullback and uniform attractors with bounds independent of the domain parameter.

Findings

Existence of solutions for each domain configuration.

Existence of pullback and uniform attractors.

Uniform bounds for attractors as the channel collapses.

Abstract

This work is devoted to the study of the asymptotic behavior of nonautonomous reaction-diffusion equations in Dumbbell domains $\Omega_{\varepsilon} \subset \mathbb{R}^{N}$. Each $\Omega_{\varepsilon}$ is the union of a fixed open set $\Omega$ and a channel $R_{\varepsilon}$ that collapses to a line segment $R_0$ as $\varepsilon \rightarrow 0^{+}$. We first establish the global existence of solution for each problem by using two properties of the parabolic equation considered, which are the positivity of the solutions and comparison results for them. We prove the existence of pullback and uniform attractors and we obtain uniform bounds (in $\varepsilon$) for them.