On the blow-up of heat flow for conformal 3-harmonic maps

Chen, Chao-Nien; Cheung, L.; Choi, Y.; Law, C.

doi:10.1090/S0002-9947-02-03054-4

On the blow-up of heat flow for conformal $3$-harmonic maps
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by Chao-Nien Chen, L. F. Cheung, Y. S. Choi and C. K. Law PDF

Trans. Amer. Math. Soc. 354 (2002), 5087-5110 Request permission

Abstract:

Using a comparison theorem, Chang, Ding, and Ye (1992) proved a finite time derivative blow-up for the heat flow of harmonic maps from $D^2$ (a unit ball in ${\mathbf R}^2$) to $S^2$ (a unit sphere in ${\mathbf R}^3$) under certain initial and boundary conditions. We generalize this result to the case of $3$-harmonic map heat flow from $D^3$ to $S^3$. In contrast to the previous case, our governing parabolic equation is quasilinear and degenerate. Technical issues such as the development of a new comparison theorem have to be resolved.

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