ASYMPTOTIC LARGE TIME BEHAVIOR OF SINGULAR SOLUTIONS OF THE FAST DIFFUSION EQUATION

We study the asymptotic large time behavior of singular solutions of the fast diffusion equation ut = Delta u(m) in (R-n \{0}) x (0,infinity) in the subcritical case 0 < m < n-2/n, n >= 3. Firstly, we prove the existence of the singular solution u of the above equation that is trapped in between self-similar solutions of the form of t(-alpha) f(i)(t(-beta)chi), i = 1,2, with the initial value u0 satisfying A(1)vertical bar chi vertical bar(-gamma) <= u0 <= A(2 vertical bar)chi vertical bar(-gamma) for some constants A(2) > A(1) > 0 and 2/1-m < gamma < n-2/m, m, where beta := 1/2-gamma(1-m), alpha := 2 beta-1/1-m, and the self-similar profile fi satisfies the elliptic equation Delta f(m) + alpha f + beta chi .del f = 0 in R-n \ {0} with lim(vertical bar chi vertical bar -> 0)vertical bar chi vertical bar(alpha/beta) f(i)(chi) = A(i) and lim(vertical bar chi vertical bar ->infinity) vertical bar chi vertical bar n-2/m f(i)(chi) = D-Ai for some constants D-Ai > 0. When 2/1-m < gamma < n, under an integrability condition on the initial value u0 of the singular solution u, we prove that the rescaled function (u) over bar (y,tau) := t(alpha)u(t(beta) y, t),tau := log t, converges to some self-similar profile f as tau -> infinity.