THE BLOW-UP PROFILE FOR A
FAST DIFFUSION EQUATION WITH A
NONLINEAR BOUNDARY CONDITION

RAÚL FERREIRA, ARTURO DE PABLO,
FERNANDO QUIRÓS, JULIO D. ROSSI

Abstract:

We study positive solutions of a fast diffusion equation in the half-line with a nonlinear boundary condition,

$$\cases{u_t = (u^m)_{xx} & $(x,t)\in {\bf R}_+\times (0,T)$,\c... ...(0,t) & $t \in(0,T)$,\cr u(x,0) = u_0(x) &$x\in{\bf R}_+$,\cr}$$

where $0<m<1$ and $p>0$ are parameters. We describe in terms of $p$ and $m$ when all solutions exist globally in time, when all solutions blow up in a finite time, and when there are both blowing up and global solutions. For blowing up solutions we find the blow-up rate and the blow-up set and we describe the asymptotic behavior close to the blow-up time $T$ in terms of a self-similar profile.