The Evolution of Singularities in Fast Diffusion Equations: Infinite-Time Blow-Down
Juan Luis Vázquez, Michael Winkler · SIAM Journal on Mathematical Analysis · 2011
This work is concerned with the fast diffusion equation $u_t= abla\cdot(u^{m-1} abla u)$ with prescribed positive data on a smoothly bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq3$, and any positive $m0$ and $\gamma>0$. It is known that in the less degenerate case where $m\in(m_c,1)$, there exist two regions with different behavior depending on $\gamma$: thus, if $\gamma 0$, whereas if $\gamma\geq n$, then the singularity persists for all times and leads to global blow-up as $t\to\infty$. Our main results show that at $m=m_c$, this line bifurcates into the two curves $\gamma=\frac{2}{1-m}$ and $\gamma=\frac{n-2}{m}$, and in the intermediate region a new type of behavior appears, namely, the phenomenon of infinite-time blow-down. More precisely, for $m\leq m_c$ we have the following: (i) if $\gamma\frac{n-2}{m}$, then u maintains its singularity at the origin for all times, and moreover $u(x,t)\to\infty$ as $t\to\infty$ for all $x\in\Omega$ (infinite-time blow-up). Furthermore, the respective borderline behaviors are clarified: It is shown that when $m