Fractional Derivatives with Respect to Time for Non-Classical Heat Problem

We consider the non-classical heat equation with Caputo fractional derivative with respect to the time variable in a bounded domain Omega subset of R+ x Rd-1 for which the energy supply depends on the heat flux on a part of the boundary S = {0} x Rd-1 with homogeneous Dirichlet boundary condition on S, the periodicity on the other parts of the boundary and an initial condition. The problem is motivated by the modeling of the temperature regulation in the medium. The existence of the solution to the problem is based on a Volterra integral of second kind in the time variable t with a parameter in Rd-1, its solution is the heat flux (y, tau) bar right arrow V (y, t) = u(x) (0, y, t) on S, which is also an additional unknown of the considered problem. We establish that a unique local solution exists and can be extended globally in time.