Two-dimensional problem for an integrodifferential equation of electrodynamics

An integrodifferential equation corresponding to the two-dimensional problem of electrodynamics with dispersion is considered. It is assumed that the electrodynamic properties of a nonconducting medium with a constant magnetic permeability and the external current are independent of the $x_3$ coordinate. In this case, the third component of the electric field vector satisfies a second-order scalar integrodifferential equation with a variable permittivity of the medium. For this equation, we study the problem of finding the spatial part of the kernel entering the integral term. This corresponds to finding the part of the permittivity that depends on the electromagnetic frequency. It is assumed that the permittivity support is contained in some compact domain $\Omega\subset\mathbb R^2$. To find this coefficient inside $\Omega$, we use information on the solution of the corresponding direct problem on the boundary of $\Omega$ on a finite time interval. An estimate for the conditional stability of the solution of the inverse problem is established under the assumption that the time interval is sufficiently large.