A HEAT CONDUCTION PROBLEM WITH SOURCES DEPENDING ON THE AVERAGE OF THE HEAT FLUX ON THE BOUNDARY

Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the do- main D = Rn-1 x R+ for which the internal energy supply depends on an average in the time variable of the heat flux (y, s) bar right arrow V(y, s) = u(x) (0, y, s) on the boundary S = partial derivative D. The solution to the problem is found for an integral representation depending on the heat flux on S which is an additional unknown of the considered problem. We obtain that the heat flux V must satisfy a Volterra integral equation of the second kind in the time variable t with a parameter in Rn-1. Under some conditions on data, we show that a unique local solution exists, which can be extended globally in time. Finally in the one-dimensional case, we obtain the explicit solution by using the Laplace transform and the Adomian decomposition method.