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.

中文翻译：

---

确定粘弹性方程的二维核的问题

正在考虑的是粘弹性的积分微分方程。直接的问题是从方程的初始边界值问题确定位移矢量的z分量。我们假设方程积分项的核取决于时间和空间变量x。为了确定核，在y = 0的直接问题的解上提出了附加条件。反问题被等效函数的积分微分方程的等价系统代替，用于未知函数。对于该系统，我们应用解析函数的Banach空间的尺度方法。反函数的局部唯一可解性在x解析的函数类中得到证明并且在t中连续。