Discontinuity Waves, Shock Formation and Critical Temperature in Crystals

The propagation of discontinuity waves in a rigid heat conductor at low temperatures is studied by using a generalized non-linear Maxwell-Cattaneo equation developed in the framework of extended thermodynamics. The critical time (i.e., the instant in which a shock wave formation occurs) is evaluated in both cases of infinite and finite heat conductivity. The critical temperature e, pointed out in our previous papers concerning the propagation of shock and simple waves, once more plays an important role: in fact, now it determines two different regimes for the wave propagation and this phenomenon, from a mathematical point of view, is related to the loss of the genuine non-linearity when fl = 6. In the last sections some numerical results are given and a brief analysis about the evolution of a possible initial wave profile is performed.