Variable instability of a constant blow-up solution in a nonlinear heat equation

This paper is concerned with positive solutions of the semilinear diffusion equation u(t) = Deltau + u(p) in Omega under the Neumann boundary condition, where p > 1 is a constant and Omega is a bounded domain in R-N with C-2 boundary. This equation has the constant solution (p - 1)(-1/(p-1))(T-0 - t)(-1/(p-1)) (0 less than or equal to t less than or equal to T-0) with the blow-up time T-0 > 0. It is shown that for any epsilon > 0 and open cone Gamma in {f is an element of C(Omega) \ f(x) > 0}, there exists a positive function u(0)(x) in Omega with partial derivativeu(0)/partial derivativev=0 on partial derivativeOmega and parallel tou(0)(x) - (p - 1)T--1/(p-1)(0)-1/(p-1) parallel to (C2(Omega)) < epsilon such that the blow-up time of the solution u(x,t) with initial data u(x, 0) = u(0)(x) is larger than T-0 and the function u(x,T-0) belongs to the cone T. A theorem on the blow-up profile is also given.