The Blow–up Rate for a Semilinear Parabolic Equation with a Nonlinear Boundary Condition

In this paper we obtain the blow-up rate for positive solutions of ut = uxx−λu, in (0, 1)×(0, T ) with boundary conditions ux(1, t) = uq(1, t), ux(0, t) = 0. If p < 2q − 1 or p = 2q − 1, 0 < λ < q, we find that the behaviour of u is given by u(1, t) ∼ (T − t) − 1 2(q−1) and, if λ < 0 and p ≥ 2q − 1, the blow up rate is given by u(1, t) ∼ (T − t) − 1 p−1 . We also characterize the blow-up profile in similarity variables.