Reconstruction of kernel depending also on space variable

The paper deals with the reconstruction of the convolution kernel, together with the solution, in a mixed linear evolution system of hyperbolic type. This problem describes uniaxial deformations u of a cylindrical domain (0, π)×Ω, which is filled with a linear viscoelastic solid whose material properties are supposed to be uniform on Ω-sections perpendicular to the x axis. Various types of boundary conditions in [0, T ] × {0, π} × Ω are prescribed, whereas Dirichlet conditions are assumed in [0, T ] × (0, π) × ∂Ω. In order to reconstruct both u and k we suppose of knowing for any time t and any x ∈ (0, π) the flux of the viscoelastic stress vector through the boundary of the Ω-section. The main novelty is that the unknown kernel k is allowed to depend, not only on the time variable t, but also on the space variable x.