Blowup behavior of solutions for a semilinear heat equation with supercritical nonlinearity

AbstractThis paper is concerned with blowup phenomena of solutions for the Cauchy and the Cauchy–Dirichlet problem of (P)ut=Δu+upwith p in the supercritical range in the sense of Sobolev's embedding. We first show that if p>1+7/(N−11) and N⩾12, then there are no radially symmetric bounded positive solutions ofΔw−y2∇w−1p−1w+wp=0inRNwhich intersect the radially symmetric singular solution at least twice. Using the above result, the existence of a blowup solution of type II for the Cauchy–Dirichlet problem for (P) in a ball is proved, where a solution u is said to exhibit the type II blowup at t=T if limsupt↗T(T−t)1/(p−1)|u(t)|∞=∞