  This paper deals with the time differential dual-phase-lag heat transfer
models aiming, at first, to identify the eventually restrictions that make them
thermodynamically consistent. At a first glance it can be observed that the
capability of a time differential dual-phase-lag model of heat conduction to
describe real phenomena depends on the properties of the differential operators
involved in the related constitutive equation. In fact, the constitutive
equation is viewed as an ordinary differential equation in terms of the heat
flux components (or in terms of the temperature gradient) and it results that,
for approximation orders greater than or equal to five, the corresponding
characteristic equation has at least a complex root having a positive real
part. That leads to a heat flux component (or temperature gradient) that grows
to infinity when the time tends to infinity and so there occur some
instabilities. Instead, when the approximation orders are lower than or equal
to four, this is not the case and there is the need to study the compatibility
with the Second Law of Thermodynamics. To this aim the related constitutive
equation is reformulated within the system of the fading memory theory, and
thus the heat flux vector is written in terms of the history of the temperature
gradient and on this basis the compatibility of the model with the
thermodynamical principles is analyzed.
