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

We study positive solutions of a fast diffusion equation in the half-line with a nonlinear boundary condition, $\left\{ {\begin{array}{*{20}{c}} {{u_t} = {{\left( {{u^m}}\right)}_{xx}}\,\left( {x,t} \right) \in {R_ + } \times \left( {0,T}\right),} \\ { - {{\left( {{u^m}} \right)}_x}\left( {0,t} \right) = {u^p}\left( {0,t} \right)\,t \in \left( {0,T} \right),} \\ {u\left( {x,0} \right) = {u_0}\left( x \right)\,x \in {R_{ + ,}}} \\ \end{array} } \right.$ 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.