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

Source abstract

A solution u u of a Cauchy problem or a Cauchy-Dirichlet problem for a semilinear heat equation $$u t = Δ u + u p u_t = \Delta u + u^p$$ with nonnegative initial data u 0 u_0 is said to undergo type II blow-up at t = T t = T if $$lim sup t ↗ T ( T − t ) 1 / ( p − 1 ) | u ( t ) | ∞ = ∞ . \limsup _{t \nearrow T} \; (T-t)^{1/(p-1)} |u(t)|_\infty = \infty .$$ Let φ ∞ \varphi _\infty be the radially symmetric singular steady state of the Cauchy problem. Suppose that u 0 ∈ L ∞ u_0 \in L^\infty is a radially symmetric function such that u 0 − φ ∞ u_0 - \varphi _\infty and ( u 0 ) t (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 u_0 which undergoes type II blow-up in the case of p > p J L p > p_{_{JL}} , where p J L p_{_{JL}} is the exponent of Joseph and Lundgren.