Diffusive stability and self-similar decay for the harmonic map heat flow.

In this paper we study the harmonic map heat flow on the euclidean space R d and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B ˙ p , ∞ d p (R d) where d < p < ∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖ ∇ u (t) ‖ L ∞ (R d) ≤ C t − 1 2 . Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity. [ABSTRACT FROM AUTHOR]