  Motivated by the modeling of temperature regulation in some mediums, we
consider the non-classical heat conduction equation in the domain
$D=\mathbb{R}^{n-1}\times\br^{+}$ for which the internal energy supply depends
on 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 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 second kind in the time
variable $t$ with a parameter in $\mathbb{R}^{n-1}$. 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.
