Growup of solutions for a semilinear heat equation with supercritical nonlinearity

We consider a Cauchy problem for a semilinear heat equation Let v∞ be the radially symmetric singular steady state of (P). It is proved that if and N 11, then for each nonnegative even integer n there exists a radially symmetric global solution un of (P) with n intersections with v∞ such that t−anun(t)∞→1 as t→∞ for some an>0 depending on n. The exact value of an is also given. We show that a0 is the optimal upper bound of growup rate for solutions below v∞.