  In this paper, we study two Energy Balance Models with Memory arising in
climatology, which consist in a 1D degenerate nonlinear parabolic equation
involving a memory term, and possibly a set-valued reaction term (of Sellers
type and of Budyko type, in the usual terminology). We provide existence and
regularity results, and obtain uniqueness and stability estimates that are
useful for the determination of the insolation function in Sellers' model with
memory.
