The problem of determining the two-dimensional kernel of a viscoelasticity equation

Under consideration is the integro-differential equation of viscoelasticity. The direct problem is to determine the $z$-component of the displacement vector from the initial boundary value problem for the equation. We assume that the kernel of the integral term of the equation depends on time and a spatial variable $x$. For determination of the kernel the additional condition is posed on the solution of the direct problem for $y=0$. The inverse problem is replaced by an equivalent system of integro-differential equations for the unknown functions. To this system, we apply the method of scales of Banach spaces of analytic functions. The local unique solvability of the inverse problem is proved in the class of functions analytic in $x$ and continuous in $t$.