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, k(r) = k(2)(k(theta theta) + 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 root(T - t)(-1). In this paper, it is proved that sup(theta) k(theta,t) approximate to root(T - t)(-1) log log(T - t)(-1) as t NE arrow 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. Velazquez. In addition, we prove some of numerical conjectures by Watterson which are keys to provide the blow-up rate. (C) 2016 Elsevier Inc. All rights reserved.