Global solvability of a kernel determination problem in 2D heat equation with memory

The inverse problem of determining two dimensional kernel in the integro-differential heat equation is considered in this paper. The kernel depends on the time variable $t$ and space variable $x$. Assuming that kernel function is given, the direct initial-boundary value problem with Neumann conditions on the boundary of a rectangular domain is studied for this equation. Using the Green's function, the direct problem is reduced to integral equation of the Volterra-type of the second kind. Then, using the method of successive approximation, the existence of a unique solution of this equation is proved. The direct problem solution on the plane $y = 0$ is used as an overdetermination condition for inverse problem. This problem is replaced by an equivalent auxiliary problem which is more suitable for further study. Then the last problem is reduced to the system of integral equations of the second order with respect to unknown functions. Applying the fixed point theorem to this system in the class of continuous in time functions with values in the Hölder spaces with exponential weight norms, the main result of the paper is proved. It consists of the global existence and uniqueness theorem for inverse problem solution.