Fast diffusion equation with critical Sobolev exponent in a ball (Q2782719)

Asymptotic extinction behaviour is studied for the fast diffusion equation in the unit ball \(\Omega=\{|x|<1\}\subset {\mathbb R}^N,\) \(N\geq 3,\) with Dirichlet's boundary conditions NEWLINE\[NEWLINE u_t=\Delta u^m\quad \text{in} \Omega,\qquad u(x,t)=0\quad \text{on} \partial\Omega\times (0,T),NEWLINE\]NEWLINE and positive symmetric initial data \(u_0(r),\) \(r=|x|.\) Precisely, let \(T\in(0,\infty)\) be the extinction time: \(u>0\) in \(\Omega\times(0,T)\) and \(u(\cdot,T)=0.\) It is shown that in the critical Sobolev case NEWLINE\[NEWLINE m=m_s\equiv {{N-2}\over{N+2}}, NEWLINE\]NEWLINE the asymptotic behaviour as \(t\to T^-\) near the origin \(x=0\) is essentially nonselfsimilar (unlike the cases \(m\in (m_s,1)\) and \(m\in (0,m_s)\)) and is constructed by matching the expansions in the inner and boundary (outer) domains. This gives the extinction rate as \(t\to T^-:\) NEWLINE\[NEWLINE \|u(\cdot,t)\|_\infty=\gamma_0 (T-t)^{(N+2)/4} |\log (T-t)|^{(N+2)/2(N-2)} (1+o(1)), NEWLINE\]NEWLINE where \(\gamma_0=\gamma_0(N)>0\) is a constant.NEWLINENEWLINENEWLINEExtinction behaviour in the supercritical case \(m\in(0,m_s)\) and another nonlinear reaction-diffusion equations with similar behaviour are also discussed.