Blow up results and localization of blow up points in a n-dimensional smooth domain

D. RIAL, J. D. ROSSI

DUKE MATHEMATICAL JOURNAL

Año: 1997 vol. 88 p. 391 - 405

0012-7094

Positive solutions of the heat equation are studied, on a bounded and smooth cylindrical domain $\Omega\times ]0,T[$, subject to the conditions $D_eta u=f(u)$ on $\partial\Omega\times ]0,T[$ and $u(\cdot,0) =u_0$ on $\Omega$. Here $ f $ is an increasing, positive, $C^2$-function such that $\int^{\infty} (1/f) $ while $u_0$ is a positive function in $C^{2+alpha} (\Omega)$ such that $D_\eta u_0 =f(u_0)$. An estimate for the blow-up time is established. Under the additional hypotheses that $\Omega= B(0,1)$, $f(0)=0$, and $f$ is convex, it is also proved that the blow-up occurs at the boundary. For a general $\Omega$, the same conclusion is shown to hold if the hypothesis $\Delta u_0\ge c>0$ is also assumed.