Some remarks on the model of rigid heat conductor with memory: unbounded heat relaxation function

TL;DR

This paper examines a heat conduction model with memory where the heat relaxation function is unbounded at initial time, revising definitions and establishing the equivalence of heat flux and thermal work under these conditions.

Contribution

It introduces a revised framework for heat flux and thermal work when the relaxation function is unbounded, highlighting the physical relevance of equivalence in this context.

Findings

Heat flux and thermal work are equivalent even with unbounded relaxation functions.

The relaxation function can be integrable but its derivative not, affecting model assumptions.

The notion of equivalence remains valid under relaxed assumptions.

Abstract

The model of rigid linear heat conductor with memory is reconsidered focussing the interest on the heat relaxation function. Thus, the definitions of heat flux and thermal work are revised to understand where changes are required when the heat flux relaxation function $k$ is assumed to be unbounded at the initial time $t=0$. That is, it is represented by a regular integrable function, namely $k\in L^1(\R^+)$, but its time derivative is not integrable, that is $\dot k\notin L^1(\R^+)$. Notably, also under these relaxed assumptions on $k$, whenever the heat flux is the same also the related thermal work is the same. Thus, also in the case under investigation, the notion of equivalence is introduced and its physical relevance is pointed out.