Finite propagation speeds in a theory of linear isotropic heat conduction

Recently a new theory of heat conduction has appeared in the literature. The raison d'etre of this theory is that in the classical theory heat propagates in a body with infinite speed. The present paper deals with the linearized form of the theory, which gives rise to an integro-partial differential equation. Two problems for this equation, called history-value problems, are posed. It is shown that, under certain conditions, solutions to these history-value problems on a bounded region of space are unique. Next, it is shown that if the data of the problem have bounded support, then for any time the solution has bounded support. This proves the hypothesis of finite wave speeds. This result is then used to prove that solutions to the history-value problems on an unbounded region of space are unique.