Blow-up profile of a solution for a nonlinear heat equation with small diffusion

This paper is concerned with positive solutions of semilinear diffusion equations u(t) = epsilon(2) Deltau + u(p) in Omega with small diffusion under the Neumann boundary condition, where p > 1 is a constant and 0 is a bounded domain in RN with C boundary. For the ordinary differential equation u(t) = u(p), the solution u(0) with positive initial data u(0) is an element of C((Omega) over bar) has a blow-up set S-0 = {x is an element of (Omega) over bar \ u(0)(x) = max (yis an element of(Omega) over bar) u(0) (y)} and a blowup profile Graphics outside the blow-up set S-0. For the diffusion equation u(t) = epsilon(2) Deltau + u(p) in Omega under the boundary condition thetau/thetav = 0 on thetaOmega, it is shown that if a positive function u(0) is an element of C-2((Omega) over bar) satisfies thetau(0)/thetav = 0 on thetaOmega, then the blow-up profile u(epsilon)/(*) (x) of the solution u(epsilon) with initial data uo approaches u(0)/(*) (x) uniformly on compact sets of (Omega) over bar \S-0 as epsilon --> +0.