Non-classical heat conduction problem with non local source
  We consider the non-classical heat conduction equation, in the domain $D=\br^{n-1}\times\br^{+}$, for which the internal energy supply depends on an integral function in the time variable of % $(y , t)\mapsto \int_{0}^{t} u_{x}(0 , y , s) ds$, %where $u_{x}(0 , y , s)$ is the heat flux on the boundary $S=\partial D$, with homogeneous Dirichlet boundary condition and an initial condition. The problem is motivated by the modeling of temperature regulation in the medium. The solution to the problem is found using a Volterra integral equation of second kind in the time variable $t$ with a parameter in $\br^{n-1}$. The solution to this Volterra equation is the heat flux $(y, s)\mapsto V(y , t)= u_{x}(0 , y , t)$ on $S$, which is an additional unknown of the considered problem. We show that a unique local solution exists, which can be extended globally in time. Finally a one-dimensional case is studied with some simplifications, we obtain the solution explicitly by using the Adomian method and we derive its properties. 
Mahdi Boukrouche and Domingo A. Tarzia
math.AP