On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity

Abstract

The inverse problem of determining a multidimensional kernel of an integral term depending on a time variable $t$ and $ (n-1)$-dimensional spatial variable $x'=\left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional heat equation with a variable coefficient of thermal conductivity is investigated. The direct problem is the Cauchy problem for this equation. The integral term has the time convolution form of kernel and direct problem solution. As additional information for solving the inverse problem, the solution of the direct problem on the hyperplane $x_n = 0$ is given. At the beginning, the properties of the solution to the direct problem are studied. For this, the problem is reduced to solving an integral equation of the second kind of Volterra-type and the method of successive approximations is applied to it. Further the stated inverse problem is reduced to two auxiliary problems, in the second one of them an unknown kernel is included in an additional condition outside integral. Then the auxiliary problems are replaced by an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the Hölder class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of the inverse problem solution.