  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.
