Ill-posed problems in thermomechanics

Michael Dreher, Ramón Quintanilla, Reinhard Racke

Research output: Contribution to journalArticle

56 Citations (Scopus)

Abstract

Several thermomechanical models have been proposed from a heuristic point of view. A mathematical analysis should help to clarify the applicability of these models, among those recent thermal or viscoelastic models. Single-phase-lag and dual-phase-lag heat conduction models can be interpreted as formal expansions of delay equations. The delay equations are shown to be ill-posed, as are the formal expansions of higher order - in contrast to lower-order expansions leading to Fourier's or Cattaneo's law. The ill-posedness is proved, showing the lack of continuous dependence on the data, and thus showing that these models (delay or higher-order expansion ones) are highly explosive. In this note we shall present conditions for when this happens. (C) 2009 Elsevier Ltd. All rights reserved.

Original languageEnglish
Pages (from-to)1374-1379
Number of pages6
JournalApplied Mathematics Letters
Volume22
Issue number9
DOIs
Publication statusPublished - Sep 2009

Keywords

  • Hyperbolic models in heat conduction
  • Stability
  • Ill-posed
  • LAG HEAT-CONDUCTION
  • STABILITY

Cite this

Dreher, M., Quintanilla, R., & Racke, R. (2009). Ill-posed problems in thermomechanics. Applied Mathematics Letters, 22(9), 1374-1379. https://doi.org/10.1016/j.aml.2009.03.010