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

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( \mathbb{R}^+) $, but its time derivative is not integrable, that is $ \dot k\notin L^1( \mathbb{R}^+) $. The study takes its origin in [2]: the heat conductor model described therein is modified in such a way to adapt it to the case of a heat flux relaxation function $ k $ which is unbounded at $ t = 0 $. Notably, also when these relaxed assumptions on $ k $ are introduced, whenever two different thermal states which correspond to the same heat flux are considered, then both states correspond also to the same thermal work. Accordingly, the notion of equivalence can be introduced, together with its physical relevance, both in the regular kernel case in [2] as well as in the singular kernel case analysed in the present investigation.