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 the parabolic integro-differential equation Dtu(t,x) = Au(t,x) + integral 0t k(t-s,|x|)Bu((s,x)ds + integral 0t D|x|) (k-s,|x|)Cu(s,x) ds+f(t,x), A being a uniformly elliptic second-order linear operator in divergence form. We single out a special class of operators 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.