The problem of determining the 2D-kernel in a system of integro-differential equations of a viscoelastic porous medium

Under consideration is the system of integro-differential equations of viscoelastic porous medium. The direct problem is to define the $y$-component of the displacement vectors of the elastic porous body and the liquid from the initial boundary value problem for these equations. We assume that the kernel of the integral term of the first equation depends on time and one of the spatial variables. To determine the kernel, some additional condition is given on the solution of the direct problem for $z=0$. The inverse problem is replaced by an equivalent system of integro-differential equations for the unknown functions. We apply the method of scales of the Banach spaces of analytic functions. The local solvability of the inverse problem is proved in the class of the functions analytic in $x$ and continuous in $t$.