A problem of identification of a special 2D memory kernel in an integro–differential hyperbolic equation

We consider an inverse problem for a partial integro–differential equation of the second order related to recovering a kernel (memory) in the integral term of this equation. It is supposed that the unknown kernel is a trigonometric polynomial with respect to the spatial variables with coefficients continuous with respect to the time variable. The direct problem for a hyperbolic integro–differential equation is the initial-boundary value problem for the half-space $x > 0$ with the zero initial Cauchy data and a special Neumann data at $x = 0$. Local existence theorem and stability estimates for the solution to the inverse problem are obtained.