Growup of solutions for a semilinear heat equation with supercritical nonlinearity

We consider a Cauchy problem for a semilinear heat equationequation(P){ut=Δu+upinRN×(0,∞),u(x,0)=u0(x)⩾0inRN. Let v∞v∞ be the radially symmetric singular steady state of (P). It is proved that if p>N−2N−1N−4−2N−1 and N⩾11N⩾11, then for each nonnegative even integer n   there exists a radially symmetric global solution unun of (P) with n   intersections with v∞v∞ such that t−an|un(t)|∞→1t−an|un(t)|∞→1 as t→∞t→∞ for some an>0an>0 depending on n  . The exact value of anan is also given. We show that a0a0 is the optimal upper bound of growup rate for solutions below v∞v∞.