Some continuous dependence results about high-order time differential thermoelastic models

ABSTRACT

This article aims to contribute to the investigation of the well-posedness question for three different heat conduction thermoelastic models, obtained starting from the dual-phase-lag (DPL) constitutive equation in its most general time differential formulation and considering Taylor expansion orders higher than those most commonly studied in literature so far. It is necessary to emphasize right from now that the investigation of such thermomechanical models, although they originate in terms of constitutive equations from suitable Taylor series expansions, has to be properly interpreted not as an attempt to emulate the delayed behavior characteristic of the original (i.e. not expanded) DPL model, but rather with the awareness of deepening the well-posedness question for three different stand-alone and self-consistent models. In particular, three estimates are obtained (one for each of the considered models) able to show the continuous dependence of the solutions of the related initial boundary value problems with respect to the supply terms and to the initial given data. All the continuous dependence results are obtained without the need to impose particular restrictions involving the delay times, except for the requirement that they are strictly positive.

KEYWORDS:

• 35G46
• 80A20
• continuous dependence
• delay times
• high-order Taylor expansion
• time differential heat conduction models

Acknowledgments

The author is very grateful to the anonymous reviewer for his careful analysis and for his valuable comments, which have led to an important improvement of the work. Moreover, the author wishes to sincerely thank Prof. Stan Chiriţă for the fruitful discussions about the topics addressed and for his helpful suggestions.