Global solution to a non-classical heat problem in the semi-space $\mathbb {R}^{+}\times \mathbb {R}^{n-1}$

We consider the non-classical heat equation in the $n$-dimensional domain $D=\mathbb {R}^{+}\times \mathbb {R}^{n-1}$ for which the internal energy supply depends on the heat flux on the boundary $S=\partial D$. 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 $\mathbb {R}^{n-1}$. 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.