About Global Solvability of a Multidimensional Inverse Problem for an Equation with Memory

Abstract

Under study is the multidimensional inverse problem of determining the convolutional kernel of the integral term in an integro-differential wave equation. The direct problem is represented by a generalized initial-boundary value problem for this equation with zero initial data and the Neumann boundary condition in the form of the Dirac delta-function. For solving the inverse problem, the traces of the solution to the direct problem on the domain boundary are given as an additional condition. The main result of the article is the theorem of global unique solvability of the inverse problem in the class of functions continuous in the time variable $ t $ and analytic in the space variable. We apply the methods of scales of Banach spaces of real analytic functions of variable and weight norms in the class of continuous functions.