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

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 R 2 ) to S 2 (a unit sphere in 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.