Parabolic integrodifferential identification problems related to radial memory kernels I

We are concerned with the problem of recovering the radial kernel $k$, depending also on time, in a parabolic integro-differential equation $$D_{t}u(t,x)={cal A}u(t,x)+int_0^t k(t-s,|x|){cal B}u(s,x)ds +int_0^t D_{|x|}k(t-s,|x|){cal C}u(s,x)ds+f(t,x),$$ ${cal A}$ being a uniformly elliptic second-order linear operator in divergence form. We single out a special class of operators ${cal A}$ and two pieces of suitable additional information for which the problem of identifying $k$ can be uniquely solved locally in time when the domain under consideration is a spherical corona or an annulus.