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
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