Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation

We consider initial-boundary value problems for a quasi linear parabolic equation, kt=k2(kθθ+k)kt=k2(kθθ+k), with zero Dirichlet boundary conditions and positive initial data. It has known that each of solutions blows up at a finite time with the rate faster than (T−t)−1. In this paper, it is proved that supθ⁡k(θ,t)≈(T−t)−1log⁡log⁡(T−t)−1 as t↗Tt↗T under some assumptions. Our strategy is based on analysis for curve shortening flows that with self-crossing brought by S.B. Angenent and J.J.L. Velázquez. In addition, we prove some of numerical conjectures by Watterson which are keys to provide the blow-up rate.