Advances on the Time Differential Three-Phase-Lag Heat Conduction Model and Major Open Issues

The main purpose of this short contribution is to summarize the recent achievements concerning the so-called time dierential three-phase-lag heat conduction model, contextually focusing attention on some of the numerous open problems associated with such an attractive theory. After having briefly recalled the origin of the model at issue, the restrictions upon the delay times and the constitutive tensors able to make it thermodynamically consistent are recalled. Under these hypotheses, the investigation of the well-posedness issue has already provided important results in terms of uniqueness and continuous dependence of the solutions (even related to the thermoelastic case), as well as in terms of existence of a domain of influence of the assigned data in connection with the thermoelastic model. Finally, some of the main problems currently object of investigation are recalled, including the very challenging issues about the dierent possible choices of Taylor series expansion orders for the constitutive equation, the interaction of the model with energy processes that take place on the nanoscale, with multi-porous materials and with biological systems.