Initial and boundary value problems of hyperbolic heat conduction

This is a study on the initial and boundary value problem of a symmetric hy perbolic system which is related to the conduction of heat in solids at low temperatures. The nonlinear system consists of a conservation equation for the energy density e and a balance equation for the heat flux Q(i), where e and Q(i) are the four basic fields of the theory. The initial and boundar y Value problem that uses exclusively prescribed boundary data for the ener gy density e is solved by a new kinetic approach that was introduced and ev aluated by Dreyer and Kunik in [1], [2] and Pertame [3]. This method includ es the formation of shock fronts and the broadening of heat pulses. These e ffects cannot be observed in the linearized theory, as it is described in [ 4]. The kinetic representations of the initial and boundary value problem revea l a peculiar phenomenon. To the solution there contribute integrals contain ing the initial fields e(0)(x), Q(0)(x) as well as integrals that need know ledge on energy and heat flux at a boundary. However, only one of these qua ntities can be controlled in an experiment. When this ambiguity is removed by continuity conditions, it turns out that after some very short time the energy density and heat flux are related to the initial data according to t he Rankine Hugoniot relation.