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

We consider the energy critical harmonic heat flow from ${\mathbb{R}}^2$ into a smooth compact revolution surface of ${\mathbb{R}}^3$. For initial data with corotational symmetry, the evolution reduces to the semilinear radially symmetric parabolic problem $${\mathrm{\partial }}_{t}u-{\text{pa}}_r^{2}u-\frac{{\text{pa}}_{r}u}{r}+\frac{f(u)}{r^2}=0$$ 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 $$\mathrm{\nabla }u(t,r)-\mathrm{\nabla }Q(\frac{r}{\text{l}(t)})\to u^{\ast }in{L}^2$$ at a speed given by the {\it quantized} rates: $$\text{l}(t)=c(u_0)(1+o(1))\frac{(T-t{)}^L}{\mathrm{\mid }\log (T-t){\mathrm{\mid }}^{\frac{2L}{2L-1}}},$$ in accordance with the formal predictions [3]. The case L=1 corresponds to the stable regime exhibited in [37], and the data for $L\ge 2$ leave on a manifold of codimension $(L-1)$ in some weak sense. Our analysis lies in the continuation of [36,32,37] by further exhibiting the mechanism for the existence of the excited slow blow up rates and the associated instability of these threshold dynamics.