Global solution to a non-classical heat problem in the semi-space r+ × rn−1

Abstract

We consider the non-classical heat equation in the n-dimensional domain D = R+ × Rn−1 for which the internal energy supply depends on the heat flux on the boundary S = ∂D. The problem is motivated by the modeling of temperature regulation in the medium. Using the Green function for the domain D, the solution is found for an integral representation depending on the heat flux V on S which is an additional unknown of the problem. We obtain that V must satisfy a Volterra integral equation of second kind at time t with a parameter in Rn−1. Under some conditions on data, we show that there exists a unique local solution which can be extended globally in time. This work generalizes the results obtained in the one-dimensional case