Trichotomy dynamics of the 1-equivariant harmonic map flow

Abstract

For the 1-equivariant harmonic map flow from $ R^2$ into $S^2$ \begin{equation*} \left\{ \begin{aligned} &v_t=v_{rr}+\frac{v_r}{r} - \frac{\sin(2v)}{2r^2} , ~\quad(r,t)\in R_+\times (t_0,+\infty),\\ &v(r,t_0)=v_0, \qquad\qquad\qquad\quad r\in R_+, \end{aligned} \right. \end{equation*} we construct global growing, bounded and decaying solutions with the initial data $v_0(r)$ satisfying $$v_0(0)=\pi ~\mbox{ and }~ v_0(r)\sim r^{1-\gamma} ~\mbox{ as }~ r\to+\infty, \quad \gamma>1.$$ These global solutions exhibit the following trichotomy long-time asymptotic behavior \begin{equation*} \| v_r(\cdot,t) \|_{L^\infty ([0,\infty))} \sim \begin{cases} t^{\frac{\gamma-2}{2}}\ln t ~&\mbox{ if }~ 1<\gamma<2,\\ 1 ~&\mbox{ if }~ \gamma=2,\\ \ln t ~&\mbox{ if }~ \gamma>2,\\ \end{cases} ~\mbox{ as }~ t\to +\infty. \end{equation*}