Fast diffusion equation with critical Sobolev exponent in a ball

We study the asymptotic extinction behaviour for the fast diusion equation in the unit ball = fjxj < 1g R N , N 3, with the Dirichlet boundary conditions u t = u m in (0; T ); u(x; t) = 0 on @ (0; T ); and positive symmetric initial data u 0 (r); r = jxj. Let T 2 (0; 1) be the extinction time: u > 0 in (0; T ) and u(; T ) = 0. We show that in the critical Sobolev case m = m s (N 2)=(N + 2) the asymptotic behaviour as t ! T near the origin x = 0 is essentially nonself-similar (unlike the cases m 2 (m s ; 1) and m 2 (0; m s )) and is constructed by matching the expansions in the inner and boundary (outer) domains. This gives the extinction rate as t ! T : ku(; t)k 1 = 0 (T t) (N+2)=4 j ln(T t)j (N+2)=2(N 2) (1 + o(1)); where 0 = 0 (N) > 0 is a constant. In addition, the extinction behaviour in the supercritical case m 2 (0; m s ), which has also not previously addressed, is noted. We discuss other nonlinear reaction-diusion (blow-up) problems...