Global solutions of the heat equation with a nonlinear boundary condition

We consider the heat equation with a nonlinear boundary condition, (P) { partial derivative(t)u = Delta u, x is an element of Omega, t>0, partial derivative(v)u = u(p), x is an element of partial derivative Omega, t>0, u(x,0) = phi(x), x is an element of Omega, where Omega = {x = (x', x(N)) is an element of R(N) : x(N) > 0}, N >= 2, partial derivative(t) =partial derivative/partial derivative(t), partial derivative(v) = -partial derivative/partial derivative x(N), p > 1+1/N, and (N - 2) p < N. In this paper we give a complete classification of the large time behaviors of the nonnegative global solutions of ( P).