On forward self-similar heat flow of harmonic maps

Abstract

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 $u_0(x)={\varphi }_0(\frac{x}{|x|}):{\mathbb{R}}^n\to N$ ($n\ge 2$), if $\Vert \nabla {\varphi }_0{\Vert }_{L^n({\mathbb{S}}^{n-1})}\le {\epsilon }_0$ 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 $C^{\infty }({\mathbb{R}}^n\times (0,\infty ))\cap C^{\frac{1}{n}}({\mathbb{R}}^n\times [0,\infty )\setminus \{(0,0)\})$.

19 pages