We consider the Cauchy problem ut = δu + δuδp-1u, x ε ℝN, t > 0, u(x, 0) = u0(x), x ε ℝN, where u0 ε C0(ℝN), the space of all continuous functions on ℝN that decay to zero at infinity, and p is supercritical in the sense that N ≥ 11 and p ≥ ((N - 2)2 - 4N + 8√N - 1)/(N - 2)(N - 10). We first examine the domain of attraction of steady states (and also of general solutions) in a class of admissible functions. In particular, we give a sharp condition on the initial function u0 so that the solution of the above problem converges to a given steady state. Then we consider the asymptotic behavior of global solutions bounded above and below by classical steady states (such solutions have compact trajectories in C 0(ℝN), under the supremum norm). Our main result reveals an interesting possibility: the solution may approach a continuum of steady states, not settling down to any particular one of them. Finally, we prove the existence of global unbounded solutions, a phenomenon that does not occur for Sobolev-subcritical exponents.