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 ${\epsilon }_0>0$ such that for any homogeneous of degree zero map $u_0(x)={\phi }_0(\frac{x}{\mathrm{\mid }x\mathrm{\mid }}):{\mathbb{R}}^n\to N$ ($n\ge 2$), if $\mathrm{\parallel }\mathrm{\nabla }{\phi }_0{\mathrm{\parallel }}_{L^n({\mathbb{S}}^{n-1})}\le {\epsilon }_0$ then there is a unique solution $u:{\mathbb{R}}^n\times (0,\mathrm{\infty })\to N$ to the heat flow of harmonic map \eqref{HF1} and \eqref{IC}, which is forward self-similar and belongs to $C^{\mathrm{\infty }}({\mathbb{R}}^n\times (0,\mathrm{\infty }))\cap C^{\frac{1}{n}}({\mathbb{R}}^n\times [0,\mathrm{\infty })\setminus \{(0,0)\})$.