Pullback dynamics for a semilinear heat equation with homogeneous Neumann boundary conditions on time-varying domains

TL;DR

This paper investigates the dynamics of a non-autonomous semilinear heat equation on time-varying domains, establishing equivalence to fixed domain problems, proving solution existence, and demonstrating the existence of pullback attractors.

Contribution

It introduces a differential geometry approach to transform the problem to a fixed domain and proves the existence of pullback attractors for the first time in this context.

Findings

Equivalence between non-autonomous problems on time-varying and fixed domains.

Local and global existence of solutions.

Existence of pullback attractors for the system.

Abstract

We are interested in studying a non-autonomous semilinear heat equation with homogeneous Neumann boundary conditions on time-varying domains. Using a differential geometry approach with coordinate transformations technique, we will show that the non-autonomous problem on a time-varying domain is equivalent, in some sense, to a non-autonomous problem on a fixed domain. Furthermore, we intend to show the local existence and uniqueness of solutions to this problem, as well as, to extend these solutions globally. Finally, we will show the existence of pullback attractors. To the best of our knowledge, results on attractors are new even for non-autonomous semilinear heat equations with homogeneous Neumann boundary conditions on time-varying domains subject to conditions with more restrictive assumptions