Long-time asymptotic behavior of the Hunter-Saxton equation

With $\bar{\partial}$-generalization of the Deift-Zhou steepest descent method, we investigate the long-time asymptotics of the solution to the Cauchy problem for the Hunter-Saxton (HS) equation \begin{eqnarray} &&u_{txx}-2\omega u_x+2u_xu_{xx}+uu_{xxx}=0,\quad x\in \mathbb{R},\ t>0,\nonumber\\ &&u(x,0)=u_0(x), \nonumber \end{eqnarray} where $u_0\in H^{3,4}(\mathbb{R})$ and $\omega>0$ is a constant. Using the new scale $(y,t)$ and a series of deformations to a Riemann-Hilbert problem associated with the Cauchy problem, we obtain the long-time asymptotic approximations of the solution $u(x,t)$ in two space-time regions: The solution of the HS equation decays as the speed of $\mathcal{O}(t^{-1/2})$ in the region $y/t >0$; While in the region $y/t<0$, the solution of the HS equation is depicted by a parabolic cylinder model with an residual error order $\mathcal{O}(t^{-1+\frac{1}{2p}})$ with $p>2$.