Parabolic integro-differential identification problems related to memory kernels with special symmetries

We are concerned with the problem of recovering the kernel k, depending on time and having a special spatial symmetry, in the parabolic integro-differential equation (1.1) and related to a domain which is union of level sets of each function k(t, ·). We single out a special class of differential operators A and two pieces of suitable additional information for which the problem of identifying k can be uniquely solved locally in time.