Mathematical analysis of high-order three-phase-lagging models

Abstract

The aim of this article is to provide an in-depth discussion about thermoelastic models able to take into account the effect of ultrafast strain of a deformable conductor coupled with a very refined behavior in terms of heat exchange, depicted through three distinct relaxation times and their related high-order effects. In particular, the well-posedness question is investigated dealing with a linear anisotropic and inhomogeneous medium, being able to prove the uniqueness as well as the continuous dependence of the solutions for suitable initial-boundary value problems. From a technical point of view, we underline that the main tools used are identifiable: i. in the introduction of an apposite integral operator that enters into the handling of the model, appropriately modifying the original initial-boundary value problem; ii. in the application of the Lagrange identity method in combination with the time-weighted function method and with an exponentially time-weighted Poincaré inequality. It is worth emphasizing that the results achieved are valid under very weak assumptions made on the thermoelastic features of the model.

Keywords:

• Continuous dependence
• deformable conductors
• high-order effects
• three-phase-lagging behavior
• uniqueness
• well-posedness

MATHEMATICS SUBJECT CLASSIFICATION:

• 74A15
• 74F05
• 74H25

Acknowledgments

V. Z. acknowledges the Italian National Group of Mathematical Physics (GNFM-INdAM) for supporting his research activity. The authors are very grateful to the anonymous reviewer for his/her useful comments and suggestions.

Disclosure statement

The authors declare that they have no conflict of interest.

Notes

1 Assuming $q\le m+1,$ the initial data ${\alpha }_{(2)}(\mathbf{x}), \dots , {\alpha }_{(q)}(\mathbf{x})$ that appear in relation (14) can be calculated from the equation (8) in terms of the other initial data assigned in (14). While for $q>m+1,$ some initial data on the time derivatives of α have to be explicitly prescribed, and in this situation the formulation of the problem requires a special attention. Anyway, this last case seems to lead to a ill-posed problem, as shown in [52]. Consequently, throughout this paper we assume that $q\le m+1$ but, for mere reasons of simplicity in mathematical calculation, we prefer to keep the above notations for the initial data ${\alpha }_{(2)}(\mathbf{x}), \dots , {\alpha }_{(q)}(\mathbf{x}),$ without writing their explicit expressions.