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.

摘要：

我们研究了半谐波映射流 $$ \casesno{ u_t=-(-\Delta)^{{1\over 2}}u+\bigg({1\over 2\pi}\ int_{\Bbb{R}}{|u(x)-u(s)|^2\over |xs|^2}ds\bigg)u&\quad {\rm in}\ \Bbb{R}\times (0,\infty),\cr u(\cdot,0)=u_0&\quad {\rm in}\ \Bbb{R}, } $$ 用于平滑函数 $u:\Bbb{R}\times [ 0,\infty)\to\Bbb{S}^1$。设$q_1,\ldots,q_k$为$\Bbb{R}$中的不同点，存在平滑初始数据$u_0$和平滑函数$\xi_j(t)\to q_j$, $0<\mu_j(t )\to 0$, 如 $t\to+\infty$, $j=1,\ldots,k$, 使得问题 (0.1) 的光滑解 $u_q$ 的形式为 $$ u_q=\omega_ \infty+\sum_{j=1}^k\Bigg(\omega\bigg({x-\xi_j(t)\over \mu_j(t)}\bigg)-\omega_\infty\Bigg)+\theta( x,t), $$ 其中 $\omega$ 是典型的最小能量半谐波映射, $\omega_\infty=\big({0\atop 1}\big)$, $\theta(x, t)\to 0$ 作为 $t\to+\infty$，均匀远离点 $q_j$。此外，参数函数 $\mu_j(t)$ 呈指数衰减至 $0$。