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.