On bounded and unbounded global solutions of a supercritical semilinear heat equation

We consider the Cauchy problem u_t = δu + δuδ^p-1u, x ε ℝ^N, t > 0, u(x, 0) = u₀(x), x ε ℝ^N, where u₀ ε C₀(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 ≥ 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(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.