Infinite time blow-up for half-harmonic map flow from R into S1

abstract:We study infinite time blow-up phenomenon for the half-harmonic map flow $$ \casesno{ u_t=-(-\Delta)^{{1\over 2}}u+\bigg({1\over 2\pi}\int_{\Bbb{R}}{|u(x)-u(s)|^2\over |x-s|^2}ds\bigg)u&\quad {\rm in}\ \Bbb{R}\times(0,\infty),\cr u(\cdot,0)=u_0&\quad {\rm in}\ \Bbb{R}, } $$ for a smooth function $u:\Bbb{R}\times [0,\infty)\to\Bbb{S}^1$. Let $q_1,\ldots,q_k$ be distinct points in $\Bbb{R}$, there exist a smooth initial datum $u_0$ and smooth functions $\xi_j(t)\to q_j$, $0<\mu_j(t)\to 0$, as $t\to+\infty$, $j=1,\ldots,k$, such that there exists a smooth solution $u_q$ of Problem (0.1) of the form $$ u_q=\omega_\infty+\sum_{j=1}^k\Bigg(\omega\bigg({x-\xi_j(t)\over \mu_j(t)}\bigg)-\omega_\infty\Bigg)+\theta(x,t), $$ where $\omega$ is the canonical least energy half-harmonic map, $\omega_\infty=\big({0\atop 1}\big)$, $\theta(x,t)\to 0$ as $t\to+\infty$, uniformly away from the points $q_j$. In addition, the parameter functions $\mu_j(t)$ decay to $0$ exponentially.