Grow up and slow decay in the critical Sobolev case

Abstract

We present conjectures on asymptotic behaviour of threshold solutions of the Cauchyproblem for a semilinear heat equation with Sobolev critical nonlinearity.The conjectures say that, depending on the decay rate of initial data andthe space dimension, thethreshold solutions may grow up, stabilize, or decay to zero as $t→∞$.The rates of grow up or decay are computed formally using matched asymptotics.