Parabolic integrodifferential identification problems related to radial memory kernels I

Abstract

We are concerned with the problem of recovering the radial kernel $k$, depending also on time, in a parabolic integro-differential equation $$D_{t}u(t,x)=\mathcal{A}u(t,x)+{\int }_0^{t}k(t-s,|x|)\mathcal{B}u(s,x)ds+{\int }_0^t{D}_{|x|}k(t-s,|x|)\mathcal{C}u(s,x)ds+f(t,x),$$ $\mathcal{A}$ being a uniformly elliptic second-order linear operator in divergence form. We single out a special class of operators $\mathcal{A}$ 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.

36 pages