Formation of singularities in heat propagation guided by second sound (Q1924449)

The author investigates the heat propagation (in one dimension), with the modified law (as opposed to that of Newton) for the heat flux namely, (i) internal energy depends only the absolute temperature \(\theta\), and (ii) the heat flux \(q\) is given by \(\tau(\theta)q_t+q= -\kappa(\theta)\theta_x\), \(\tau\) and \(\kappa\) are positive functions, (Cattaneo's law), which makes the system hyperbolic. It is shown that, in this case, the classical solutions with large initial data blow up in finite time, due to the formation of shock waves. The paper begins with the statement of a local existence theorem, followed by the main result. This is based on a lemma which gives a pointwise upper bound on the solution in terms of the initial data.