Kernel determination problem for one parabolic equation with memory

This paper studies the inverse problem of determining a multidimensional kernel function of an integral term which depends on the time variable $t$ and $ (n-1) $-dimensional space variable $x'= \left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional diffusion equation with a time-variable coefficient at the Laplacian of a direct problem solution. Given a known kernel function, a Cauchy problem is investigated as a direct problem. The integral term in the equation has convolution form: the kernel function is multiplied by a solution of the direct problem's elliptic operator. As an overdetermination condition, the result of the direct question on the hyperplane $x_n = 0$ is used. An inverse question is replaced by an auxiliary one, which is more suitable for further investigation. After that, the last problem is reduced to an equivalent system of Volterra-type integral equations of the second order with respect to unknown functions. Applying the fixed point theorem to this system in Hölder spaces, we prove the main result of the paper, which is a local existence and uniqueness theorem.