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

In this paper positive solutions of the heat equation with a nonlinear Neum ann boundary conditions in an upper halfspace are studied. The optimal resu lt on blow-up rate, valid for all solutions which blow up in finite time, i s established under the assumption that the exponent of a nonlinear boundar y condition is subcritical in the Sobolev sense. This complements results d erived for the bounded domain case in [10, 13] either for monotonous soluti ons or under a stronger restriction on the exponent of a boundary condition . Copyright (C) 2000 John Wiley & Sons, Ltd.