Asymptotics of the Coleman-Gurtin model

TL;DR

This paper studies a heat conduction model with hereditary memory, proving the existence of global and exponential attractors with optimal regularity and finite fractal dimension for the solution semigroup.

Contribution

It introduces a novel analysis within the history space framework for the Coleman-Gurtin model, establishing attractors with optimal regularity and finite fractal dimension.

Findings

Existence of global attractors in weak-energy and regular spaces.

Existence of exponential attractors with finite fractal dimension.

Analysis applicable to nonlinearities of critical growth.

Abstract

This paper is concerned with the integrodifferential equation $$\partial_t u-\Delta u -\int_0^\infty \kappa(s)\Delta u(t-s)\,\d s + \varphi(u)=f$$ arising in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in presence of a nonlinearity $\varphi$ of critical growth. Rephrasing the equation within the history space framework, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related solution semigroup, acting both on the basic weak-energy space and on a more regular phase space.