Problem of Determining the Two-Dimensional Kernel of the Viscoelasticity Equation with a Weakly Horizontal Inhomogeneity

In a domain bounded with respect to the variable $$ z $$ and having a weakly horizontal inhomogeneity, we consider the problem of determining the convolution kernel $$ k(t,x) $$ , $$ t\in [0,T] $$ , $$ x\in {\mathbb R} $$ , occurring in the viscoelasticity equation. It is assumed that this kernel weakly depends on the variable $$ x $$ and has a power series expansion in a small parameter $$ \varepsilon $$ . A method is constructed for finding the first two coefficients $$ k_{0}(t) $$ and $$ k_{1 }(t) $$ of this expansion. Theorems on the global unique solvability of the problem are obtained.

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