We consider the  Cauchy problem
$$
     u_t = \Delta u+|u|^{p-1}u,       x \in \R^N,   t>0,       \\
     u(x,0) = u_0(x),                       x \in \R^N,
$$
where $u_0 \in  C_0(\R^N)$, the space of all continuous functions on $\R^N$
that decay to zero at infinity, and $p$ is supercritical in the sense
that $N\ge 11$ and $p\ge ((N-2)^2-4N+8\sqrt{N-1})/{(N-2)(N-10)}$.
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 $u_0$ 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(\R^N)$,
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.