Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow

Abstract

We consider the energy-critical harmonic heat flow from ${ℝ}^2$ into a smooth compact revolution surface of ${ℝ}^3$. For initial data with corotational symmetry, the evolution reduces to the semilinear radially symmetric parabolic problem

$${\partial }_{t}u-{\partial }_r^{2}u-\frac{{\partial }_{r}u}{r}+\frac{f\left(u\right)}{r^2}=0$$

for a suitable class of functions $f$. Given an integer $L\in {\mathbb{N}}^{\ast }$, we exhibit a set of initial data arbitrarily close to the least energy harmonic map $Q$ in the energy-critical topology such that the corresponding solution blows up in finite time by concentrating its energy

$$\nabla u\left(t,r\right)-\nabla Q\left(\frac{r}{\left(t\right)}\right)\to u^{\ast }\text{ in }{L}^2$$

at a speed given by the quantized rates

$$\left(t\right)=c\left(u_0\right)\left(1+o\left(1\right)\right)\frac{{\left(T-t\right)}^L}{|log\left(T-t\right){|}^{2L∕\left(2L-1\right)}},$$

in accordance with the formal predictions of van den Berg et al. (2003). The case $L=1$ corresponds to the stable regime exhibited in our previous work (CPAM, 2013), and the data for $L\ge 2$ leave on a manifold of codimension $L-1$ in some weak sense. Our analysis is a continuation of work by Merle, Rodnianski, and the authors (in various combinations) and it further exhibits the mechanism for the existence of the excited slow blow-up rates and the associated instability of these threshold dynamics.