When considering equation (1.1) in a bounded subset Ω ⊂ R N , in view of H1, since ρ is allowed either to vanish or to diverge at S, it is natural to consider the following initial value problem associated with (1.1):
(1.5) ρ∂ t u = ∆[G(u)] in Ω × (0, T ], u = u 0 in Ω × {0},
where no boundary conditions are specified at S. We require ρ, G and f to satisfy hypotheses H1-2-3; furthermore, for the initial datum u 0 we assume that
H3. u 0 ∈ L ∞ (Ω) ∩ C(Ω).
Concerning the existence and uniqueness of the solutions to (1.5), the case G(u) = u has been largely investigated, using both analytical and stochastic methods (see, e.g., [23,28,29,32]). Also analogous elliptic or elliptic-parabolic equations have attracted much attention in the literature (see, e.g., [6,7,8,9,10,11,26,27]); in particular, the question of prescribing continuous **data** at S has been addressed (see, e.g., [23,27,28,29]).