A Diffusive-Hyperbolic Model for Heat Coduction

The present paper faces the problem of heat conduction within the framework of thermodynamics with internal state variables. A model, in which the heat flux vector depends both on the gradient of the absolute temperature and the gradient of a scalar internal variable, is proposed. Such a model leads to a diffusive-hyperbolic system which in general is parabolic, but also allows to shift to the hyperbolic regime. In the hyperbolic case the propagation of weak discontinuity waves is investigated. The Rankine-Hugoniot and Lax conditions for the propagation of strong shock waves are analyzed as well.