Existence of expanders of the harmonic map flow

If u0 is invariant under the above scaling, i.e. if u0 is 0-homogeneous, solutions of the harmonic map flow which are invariant under scaling are potentially well-suited for smoothing out u0 instantaneously. Such solutions are called expanding solutions or expanders. In this setting, it turns out that (1) is equivalent to a static equation, i.e. that it does not depend on time anymore. Indeed, if u is an expanding solution in the previous sense then the map U(x) := u(x, 1) for x ∈ Rn, satisfies the elliptic system