Abstract

Under consideration is the system of integro-differential equations of a 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 \(t \).

中文翻译：

---

粘弹性多孔介质积分微分方程组中二维核的确定问题

摘要

正在考虑的是粘弹性多孔介质的积分微分方程组。直接的问题是从这些方程式的初始边界值问题定义弹性多孔体和液体的位移矢量的\（y \）分量。我们假设第一方程积分项的核取决于时间和空间变量之一。为了确定内核，对\（z = 0 \）的直接问题的解给出了一些附加条件。反问题由未知函数的等效积分微分方程组取代。我们应用解析函数的Banach空间的尺度方法。反函数的局部可解性在\（x \）解析的函数类别和\（t \）连续 的函数类别中得到证明。