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 ${\mathbb{R}}^d$ and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space ${\dot{B}}_{p,\mathrm{\infty }}^{\frac{d}{p}}({\mathbb{R}}^d)$ where $d<p<\mathrm{\infty }$. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form $\mathrm{\parallel }\mathrm{\nabla }u(t){\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^d)}\le C{t}^{-\frac{1}{2}}$. 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.