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

摘要

We consider the Cauchy problem, where N>2, p>1, and u 0 is a bounded continuous non-negative function in R ~N. We study the case where u _0(x) decays at the rate |x| ~(-2/(p-1)) as |x|~(→∞), and investigate the stability and instability properties of forward self-similar solutions. In particular, we obtain optimal conditions on the initial function u _0 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 u _0(x) as |x|~(→∞).