A SIMPLIFIED APPROACH TO THE REFINED BLOWUP BEHAVIOR FOR THE NONLINEAR HEAT EQUATION
Résumé
We consider radial decreasing solutions of the semilinear heat equation, both for subcrit-ical and supercritical powers. We provide a much simpler and more pedagogical proof of the classical results on the sharp final blowup profile and on the refined space-time behavior in the subcritical case. We also improve some of the known results, by providing estimates in a more global form. In particular, we obtain the rate of approach of the solution to its singular final profile, given by
$$ u(x,t)\sim \biggl[(p-1)(T-t)+{(p-1)^2\over 8p}{|x|^2\over |\log |x||}\biggr]^{-1/(p-1)},
\qquad\hbox{for all $t\in [T-|x|^2,T)$,}$$
for fixed $x$ small.
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