On forward self-similar heat flow of harmonic maps

For any $k$-dimensional smooth, compact Riemannian manifold $(N, h)\subset\mathbb R^L$ without boundary, there exists an $\varepsilon_0>0$ such that for any homogeneous of degree zero map $u_0(x)=\phi_0(\frac{x}{|x|}):\mathbb R^n\to N$ ($n\ge 2$), if $\|\nabla\phi_0\|_{L^n(\mathbb S^{n-1})}\le\varepsilon_0$ then there is a unique solution $u:\mathbb R^n\times (0,\infty)\to N$ to the heat flow of harmonic map \eqref{HF1} and \eqref{IC}, which is forward self-similar and belongs to $C^\infty(\R^n\times (0,\infty))\cap C^{\frac1{n}}(\R^n\times [0,\infty)\setminus \{(0,0)\})$.