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arXiv · 2104.00408

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

Shi-Zhong Du
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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 =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.

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Shi-Zhong Du. 2021-04-01. Finite time blowup and type II rate for harmonic heat flow from Riemannian manifolds. https://arxiv.org/abs/2104.00408

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