Existence and uniqueness of global positive solutions to the degenerate parabolic problem u_t = f(u)\Delta u \ \ \mathrm {in} \ \ \mathbb R^n \times (0, \infty) u|_{t=0} = u–0 with f \in C^0 ((0, \infty)) \cap C^1 ((0, \infty)) satisfying f(0) = 0 and f(s) > 0 for s > 0 are investigated. It is proved that, without any further conditions on f , decay of u_0 in space implies uniform zero convergence of u(t) as t \rightarrow \infty . Furthermore, for a certain class of functions f explicit decay rates are established.