A nonlocal heat conduction problem

Abstract

We develop a model describing the temperature of a heated wire contained in a vessel. This containing vessel reradiates the heat received, maintaining itself at a steady-state temperature that is not a przori known but whose determination is part of the problem. The resulting Dirichlet problem involves the heat flux through the ends of the wire and the integral of a function of the temperature of the wire; it is thus nonlocal. Under physically motivated hypotheses, we prove that this problem has a unique solution.