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

Abstract

Let $(M,g)$ be a m dimensional Riemannian manifold with metric g and $(N,h)$ be a n dimensional Riemannian sub-manifold of $R^k$ with induced metric h. In this paper, we will study the existence of finite time singularity to harmonic heat flow$u_t-{△}_{g}u=A_u(\nabla u,\nabla u)$ and their formation patterns.

After works of Coron-Ghidaglia [10], Ding [11] and Chen-Ding [5], one knows blow-up solutions under smallness of initial energy for $m\ge 3$. Soon later, 2 dimensional blowup solutions were found by Chang-Ding-Ye in [8]. The first part of this paper is devoted to construction of new examples of finite time blow-up solutions without smallness conditions for $3\le m<7$. 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 $0<b<\frac{\pi }{2}$. This result can be regarded as a generalization of results of Chang-Ding-Ye [8] and Chang-Ding [6] 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 ∂M, Hamilton (mentioned by Chang-Ding-Ye in [8]) has obtained that the blowup rate must be faster than ${(T-t)}^{-1}$. Under a similar setting, it was later improved a little by Topping [23] to ${(T-t)}^{-1}|\log (T-t\left)\right|$. In this paper, we will extend the results to all Riemannian surfaces M and improve the rate of Topping to ${(T-t)}^{-1}a\left(\right|\log (T-t)\left|\right)$ for any positive nondecreasing function $a\left(\tau \right)$ satisfying$\underset{1}{\overset{\infty }{\int }}\frac{d\tau }{a\left(\tau \right)}=+\infty ,$ which is comparable to a recent result of Raphaël-Schweyer [20] for rotational symmetric solutions. Turning to the higher dimensional case $3\le m<7$, 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 result of $m\ge 7$ by Bizoń-Wasserman [4]. 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 $3\le m<7$ and degree is no less than 2.