On the blow-up rate for the heat equation with a nonlinear boundary condition

Abstract

In this paper positive solutions of the heat equation with a nonlinear Neumann boundary conditions in an upper halfspace are studied. The optimal result on blow-up rate, valid for all solutions which blow up in finite time, is established under the assumption that the exponent of a nonlinear boundary condition is subcritical in the Sobolev sense. This complements results derived for the bounded domain case in [10, 13] either for monotonous solutions or under a stronger restriction on the exponent of a boundary condition. Copyright © 2000 John Wiley & Sons, Ltd.