Hyperbolic Relaxation of Reaction Diffusion Equations with Dynamic Boundary Conditions

TL;DR

This paper investigates the hyperbolic relaxation of reaction-diffusion equations with dynamic boundary conditions, analyzing the behavior of solutions and attractors as the relaxation parameter approaches zero.

Contribution

It introduces a unified framework for hyperbolic and parabolic problems, proving the upper-semicontinuity of global attractors and establishing exponential attractors.

Findings

Existence of global attractors with optimal regularity

Upper-semicontinuity of attractors as relaxation parameter tends to zero

Existence of exponential attractors

Abstract

Under consideration is the hyperbolic relaxation of a semilinear reaction-diffusion equation on a bounded domain, subject to a dynamic boundary condition. We also consider the limit parabolic problem with the same dynamic boundary condition. Each problem is well-posed in a suitable phase space where the global weak solutions generate a Lipschitz continuous semiflow which admits a bounded absorbing set. We prove the existence of a family of global attractors of optimal regularity. After fitting both problems into a common framework, a proof of the upper-semicontinuity of the family of global attractors is given as the relaxation parameter goes to zero. Finally, we also establish the existence of exponential attractors.