A SIMPLIFIED APPROACH TO THE REFINED BLOWUP BEHAVIOR FOR THE NONLINEAR HEAT EQUATION

Philippe Souplet (1)

1 Université Sorbonne Paris Nord

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.

Domaines: Equations aux dérivées partielles [math.AP]

Philippe Souplet. A SIMPLIFIED APPROACH TO THE REFINED BLOWUP BEHAVIOR FOR THE NONLINEAR HEAT EQUATION. 2018. ⟨hal-01721261⟩