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 \(u_0(x)=\phi _0(\frac{x}{|x|}):{\mathbb {R}}^n\rightarrow N\) ( \(n\ge 2\) ), if \(\Vert \nabla \phi _0\Vert _{L^n({\mathbb {S}}^{n-1})}\le \varepsilon _0\) then there is a unique solution \(u:{\mathbb {R}}^n\times (0,\infty )\rightarrow N\) to the heat flow of harmonic map (1.1) and (1.2), which is forward self-similar and belongs to \(C^\infty ({{\mathbb {R}}}^n\times (0,\infty ))\cap C^{\frac{1}{n}}({{\mathbb {R}}}^n\times [0,\infty )\setminus \{(0,0)\})\)
