Blow-up rate of type II and the braid group theory

Abstract:

A solution $u$ of a Cauchy problem or a Cauchy-Dirichlet problem for a semilinear heat equation \[ u_t = \Delta u + u^p \] with nonnegative initial data $u_0$ is said to undergo type II blow-up at $t = T$ if \[ \limsup _{t \nearrow T} \; (T-t)^{1/(p-1)} |u(t)|_\infty = \infty . \] Let $\varphi _\infty$ be the radially symmetric singular steady state of the Cauchy problem. Suppose that $u_0 \in L^\infty$ is a radially symmetric function such that $u_0 - \varphi _\infty$ and $(u_0)_t$ change sign at most finitely many times. By application of the braid group theory, we determine the exact blow-up rate of solution with initial data $u_0$ which undergoes type II blow-up in the case of $p > p_{_{JL}}$, where $p_{_{JL}}$ is the exponent of Joseph and Lundgren.