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 $k\in L^1(\R^+)$, 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.