Harmonic map heat flow for axially symmetric data (Q1191497)

For the study of the existence of harmonic maps by heat flow methods, whether blow up occurs or not is very important. The author examines the harmonic map heat flow for maps between the three-dimensional ball and the two-sphere. It is proved that blow up can occur for certain smooth initial data. In contrast, the author proves that, given sufficiently regular, axially symmetric nonsurjective initial and boundary data, blow up does not occur and a smooth harmonic map is obtained. Generalization to higher dimensions are also discussed.