Global existence and blow-up phenomena for the periodic Hunter-Saxton equation parametrized by the speed \(\kappa \) of the Galilean frame
\textit{J. K. Hunter} and \textit{R. Saxton} [SIAM J. Appl. Math. 51, No. 6, 1498--1521 (1991; Zbl 0761.35063)] proposed a nonlinear wave equation showing that weakly nonlinear waves are described by the equation \[ (u_t+ uu_x)_x= {1\over 2} u^2_x, \] where \(u(t,x)\) describes the director field of a nematic liquid crystal, \(x\) is a space variable in a reference frame moving with the linearized wave velocity, and \(t\) is a slow time variable. In this paper, the periodic Hunter-Saxton equation parametrized by the speed \(\kappa\) of the Galilean frame is under consideration. Firstly, the local existence and the blow-up scenario of strong solutions to the equation are obtained. After that, two blow-up results are established and the blow-up rates of strong solutions to the equation are presented. Finally, it is proved that the equation also has global solutions.
scientific article; zbMATH DE number 5940420