Fast diffusion equation with critical Sobolev exponent in a ball
Victor A. Galaktionov, John R. King · Nonlinearity · 2001
We study the asymptotic extinction behaviour for the fast diffusion equation in the unit ball Ω = {| x |0 in Ω × (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 non-self-similar (unlike the cases m ∊( m s ,1) and m ∊(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 - : \| u (·, t )|| ∞ = γ 0 ( T - t ) ( N + 2)/4 |ln ( T - t )| ( N + 2)/2( N -2) (1 + o(1)), where γ 0 = γ 0 ( N )>0 is a constant. In addition, the extinction behaviour in the supercritical case m ∊(0, m s ), which has also not been previously addressed, is noted. We discuss other nonlinear reaction-diffusion (blow-up) problems admitting such a critical asymptotic behaviour.