Complexity of asymptotic behavior of the porous medium equation in RN

In this paper, we consider the complexity of large time behavior of solutions to the porous medium equation u(1) - Delta u(m) = 0 in R-N with m > 1. We first show that for any given 0 < mu < 2N/N(m-1)+2 and beta > 2-mu(m-1)/4, the omega-limit set of t mu/2 u(t(beta), t) includes all of the nonnegative functions f in the Schwartz space S(R-N) with f(0) = 0. Furthermore, we prove that, for a given countable subset E of the interval (0, 2N/(N(m-1)+2(2+mu(m-1))), there exists an initial value u(0)(x) such that for all mu and beta satisfying 0 < mu < 2N/N(m-1)+2, beta > 2-mu(m-1)/4 and mu/2 beta is an element of E, the omega-limit set of t(mu/2) u(t(beta)., t) is equal to C-0(+)(R-N) equivalent to {f is an element of C-0(R-N); f(x) >= 0, f(0) = 0}.