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Motivated by the modeling of temperature regulation in some mediums, weconsider the non-classical heat conduction equation in the domain$D=\mathbb{R}^{n-1}\times\br^{+}$ for which the internal energy supply dependson an average in the time variable of the heat flux $(y, s)\mapsto V(y,s)=u_{x}(0 , y , s)$ on the boundary $S=\partial D$. The solution to the problemis found for an integral representation depending on the heat flux on $S$ whichis an additional unknown of the considered problem. We obtain that the heatflux $V$ must satisfy a Volterra integral equation of second kind in the timevariable $t$ with a parameter in $\mathbb{R}^{n-1}$. Under some conditions ondata, we show that a unique local solution exists, which can be extendedglobally in time. Finally in the one-dimensional case, we obtain the explicitsolution by using the Laplace transform and the Adomian decomposition method.

A Heat Conduction Problem with Sources Depending on the Average of the Heat Flux on the Boundary