Finite time blowup and type II rate for harmonic heat flow from Riemannian manifolds

Abstract: In this paper, we will study the existence of finite time singularity to harmonic heat flow and their formation patterns. After works of Coron-Ghidaglia, Ding and Chen-Ding, one knows blow-up solutions under smallness of initial energy for m>=3. soon later, 2 dimensional blowup solutions were found by Chang-Ding-Ye. The first part of this paper is devoted to construction of new examples of finite time blow-up solutions without smallness conditions for 3<=m<7. In fact, when considering rotational symmetric harmonic heat flow from B_1\subset R^m to S^m\subset R^{m+1}, we will prove that the maximal solution blows up in finite time if b>\vartheta_m, and exists for all time if 0<b<\pi/2. This result can be regarded as a generalization of results of Chang-Ding-Ye nad Chang-Ding to higher dimensional case, which relies on a completely different argument. The second part of the paper study the rate of blow-up solutions. When M is a bounded domain in R^2 and consider Dirichlet boundary condition on \partial M, Hamilton has obtained that the blowup rate must be faster than (T-t)^{-1}. Under a similar setting, it was later improved a litttle by Topping to (T-t)^{-1}|log(T-t)|. In this paper, we will extend the results to all Riemmanian surfaces M and improve the rate of Topping to (T-t)^{-1}a(|log(T-t)|) for any positive nondecreasing function a(\tau) satisfying $\int^\infty_1\frac{d\tau}{a(\tau)}=+\infty$, which is comparable to a recent result of Raphael-Schweyer for rotational symmetric solutions. Turning to the higher dimensional case 3<=m\<7, we will demonstrate a completely different phenomenon by showing that all rotational symmetric blow-up solutions can not be type II, which is different to the case m>=7 by Bizon-Wasserman. Finally, we also present result of finite time type I blowup for heat flow from S^m to S^m\subset R^{m+1}, when 3<=m<7 and degree is no less than 2.