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

Abstract

We consider the energy critical harmonic heat flow from $\Bbb R^2$ into a smooth compact revolution surface of $\Bbb R^3$. For initial data with corotational symmetry, the evolution reduces to the semilinear radially symmetric parabolic problem $$\partial_t u -\pa^2_{r} u-\frac{\pa_r u}{r} + \frac{f(u)}{r^2}=0$$ for a suitable class of functions $f$ . Given an integer $L\in \Bbb N^*$, 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 $$\nabla u(t,r)-\nabla Q(\frac{r}{\l(t)})\to u^* in L^2$$ at a speed given by the {\it quantized} rates: $$\l(t)=c(u_0)(1+o(1))\frac{(T-t)^L}{|\log (T-t)|^{\frac{2L}{2L-1}}},$$ 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.