Nonuniqueness for the traveling wave speed for harmonic heat flow

Given any wave speed c (independently of its sign!), we construct a traveling wave solution of the harmonic heat flow with values in the unit sphere and defined in an infinitely long cylinder. The wave connects two locally stable and axially symmetric steady states at + and - infinity, and has a singular point on the cylinder axis. In view of the bistable character of the potential, the result is surprising, and it is intimately related to the nonuniqueness of the harmonic map flow itself. The result is counter-intuitive if c has the "wrong sign", i.e. if the steady state with higher energy invades the cylinder. We show that for only one wave speed the traveling wave behaves locally, near its singular point, as a symmetric harmonic map.