Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition

Abstract

In this paper we study the large time behavior of positive solutions of the heat equation under the nonlinear boundary condition $\text{∂u}\text{∂ν}\text{= f(u)}$, where η is the outward normal and f is nondecreasing with f(u) > 0 for u > 0. We show that if $\text{Ω = B}{}_{\mathrm{R}}$ and 1/f is integrable at infinity there is finite time blow up for any initial datum. In the two dimensional case we show that this is true for any smooth simply connected domain. In the radially symmetric case if f ϵ C² is convex and satisfies the properties above we show that blow up occurs only at the boundary.