Weak Exponential Attractors for Coleman--Gurtin Equations with Dynamic Boundary Conditions Possessing Different Memory Kernels

TL;DR

This paper extends the analysis of Coleman--Gurtin equations with dynamic boundary conditions by establishing the existence of a weak exponential attractor when the interior and boundary memory kernels differ, advancing understanding of their long-term behavior.

Contribution

It introduces the concept of a weak exponential attractor for cases where the memory kernels do not coincide, completing the analysis of finite dimensional attractors for this class of equations.

Findings

Existence of a weak exponential attractor with full basin of attraction.

The attractor is finite dimensional and compact in the weak topology.

Advances understanding of asymptotic behavior when memory kernels differ.

Abstract

The well-posedness of a generalized Coleman--Gurtin equation equipped with dynamic boundary conditions with memory was recently established by C.G. Gal and the author. Additionally, it was established by the author that the problem admits a finite dimensional global attractor and a robust family of exponential attractors in the case where singularly perturbed memory kernels defined on the interior of the domain and on the boundary of the domain coincide. In the present article we report advances concerning the asymptotic behavior of this heat transfer model when the memory kernels do {\em not} coincide. In this setting we obtain a weak exponential attractor whose basin of attraction is the entire phase space, that is, a finite dimensional exponentially attracting compact set in the weak topology of the phase space. This result completes an analysis of the finite dimensional attractors for the generalized Coleman--Gurtin equation equipped with dynamic boundary conditions with memory.