Linearized two-dimensional inverse problem of determining the kernel of the viscoelasticity equation

A linearized inverse problem of determining the 2D convolutional kernel of the integral term in an integro-differential viscoelasticity equation is considered. 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. The unknown kernel is decomposed into two components, one of which is a small in absolute value unknown additive. For solving the inverse problem, the traces of the solution to the direct problem on the domain boundary are given as an additional condition. It is proved that the linearized problem of determining the convolutional kernel is equivalent to a system of linear Volterra type integral equations. The generalized contraction mapping principle is applied. The main result of the article is the theorem of global unique solvability of the inverse problem in the class of continuous functions. A theorem on the convergence of a regularized family of problems to the solution of the original (ill-posed) problem is presented.