  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
$\dot{B}^{\frac{d}{p}}_{p,\infty}(\mathbb{R}^d)$ where $d<p<\infty$. As a
consequence we obtain decay rates for solutions of the harmonic map flow of the
form $\|\nabla u(t) \|_{L^\infty(\mathbb{R}^d)}\leq Ct^{-\frac12}$.
  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.
