The local solvability of a problem of determining the spatial part of a multidimensional kernel in the integro-differential equation of hyperbolic type

The multidimensional inverse problem of determining spatial part of integral member kernel in integro-differential wave equation is considered. Herein, the direct problem is represented by the initial-boundary problem for this with zero initial data and Neyman’s boundary condition as Dirac’s delta-function concentrated on the boundary of the domain $(x, t) \in \mathbb R^{n+1}$, $z > 0$. As information in order to solve the inverse problem on the boundary of the considered domain the traces of direct problem solution are given. The significant moment of the problem setup is such a circumstance that all given functions are real analytical functions of variables $x \in \mathbb R^n$. The main result of the work is concluded in obtaining the local unique solvability of the inverse problem in the class of continuous functions on variable $z$ and analytical on other spatial variables. For this, by means of singularity separation method, the inverse problem is replaced by the initial-boundary problem for the regular part of the solution of this problem. Further, direct and inverse problems are reduced to the solution of equivalent system of Volterra type integro-differential equations. For the solution of the latter, the method of Banach space scale of real analytical functions is used.

integro-differential equation, inverse problem, uniqueness, estimate of stability, pulse source, characteristic