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

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 ( ∇ u , ∇ 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 ≥ 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 ≤ m < 7 . In fact, when considering rotational symmetric harmonic heat flow from B 1 ⊂ R m to S m ⊂ R m + 1 , we will prove that the maximal solution blows up in finite time if b > ϑ m , and exists for all time if 0 < b < π 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 ) | . In this paper, we will extend the results to all Riemannian surfaces M and improve the rate of Topping to ( T − t ) − 1 a ( | log ⁡ ( T − t ) | ) for any positive nondecreasing function a ( τ ) satisfying ∫ 1 ∞ d τ a ( τ ) = + ∞ , which is comparable to a recent result of Raphaël-Schweyer [20] 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 result of m ≥ 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 ⊂ R m + 1 , when 3 ≤ m < 7 and degree is no less than 2.