Global existence and blow-up for harmonic map heat flow

For degree-one equivariant maps on bounded domains, the question of finite-time blow-up vs. global existence of solutions to the harmonic map heat flow has been well studied. In this paper we study the Cauchy problem for degree-m equivariant harmonic map heat flow from (2+1)-dimensional space–time into the 2-sphere with initial energy close to the energy of harmonic maps. It is proved that solutions are globally smooth for m⩾4, whereas for m=1, we show that finite-time singularities can form for this class of data.