Rigid heat conduction with memory: some recent results

Materials whose mechanical and/or thermodynamical behaviour is determined not only by their present status but also by their past history can be termed materials with memory. A well known example of material with memory is a rigid heat conductor with memory. This model, according to [8] and [1], describes a body, assumed rigid, in which the memory affects its thermodynamical behaviour. Specifically, the heat flux relaxation function depends only on the time variable through the present time as well as the whole past history. Various problems are considered both in the case of a regular as well as of singular kernel problems are considered [3, 7, 4, 5]. The term regular is adopted to indicate a heat flux relaxation function which is integrable together with its time derivative on any interval (0,T) ∈ R+; while the term singular denotes a heat flux relaxation function which is integrable on any interval (0,T) ∈ R+, its time derivative is not. Analogies between rigid heat conduction and viscoelasticity problems are also mentioned [2, 6].