2D kernel identification problem in viscoelasticity equation with a weakly horizontal homogeneity

We consider the problem of determining the kernel $k(t,x)$, $t\in [0,T]$, $x\in {\Bbb R}$, entering the equation of viscoelasticity in a bounded domain with respect to $z$ with weakly horizontal homogeneity. It is assumed that this kernel weakly depends on the variable $x$ and decomposes into a power series by degrees of the small parameter $\varepsilon$. A method for finding unknown functions $k_{0}$, $k_{1}$ is constructed. The global uniquely solvability and stability theorems are obtained.