Ill-posed problems in thermomechanics

Abstract

In the literature there exist several thermomechanical models which are proposed from a heuristic point of view. A mathematical analysis should help to clarify the applicability of these models. In recent years several thermal or viscoelastic models have been proposed in which the relaxation time or the delay time plays an important role. Single- 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 well as 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, thus showing that these models (delay, or higher-order expansions) are highly explosive. In this note we shall present conditions when this happens.