Identification of memory kernels depending on time and on an angular variable

We deal with the problem of recovering a memory kernel k(t, η), depending on time t and on an angular variable η, in a parabolic integrodifferential equation related to a toric domain. We show that the problem can be uniquely solved locally in time if the kernel k is not assumed to be necessarily periodic with respect to η. On the contrary, under a periodicity condition for k(t, ·), we show uniqueness assuming existence.