The role of forward self-similar solutions in the Cauchy problem for semilinear heat equations

We consider the Cauchy problem{ut=Δu+up,x∈RN,t>0,u(x,0)=u0(x),x∈RN, where N>2N>2, p>1p>1, and u0u0 is a bounded continuous non-negative function in RNRN. We study the case where u0(x)u0(x) decays at the rate |x|−2/(p−1)|x|−2/(p−1) as |x|→∞|x|→∞, and investigate the stability and instability properties of forward self-similar solutions. In particular, we obtain optimal conditions on the initial function u0u0 for the global existence in terms of self-similar solutions, and show the asymptotically self-similar behavior of the global solutions. We also obtain the condition for finite time blow-up by making use of the behavior of u0(x)u0(x) as |x|→∞|x|→∞.