Global existence and blow-up phenomena for the periodic Hunter–Saxton equation parametrized by the speed κκ of the Galilean frame

In this paper, we study the periodic Hunter–Saxton equation parametrized by the speed κκ of the Galilean frame. First, we obtain the local existence and the blow-up scenario of strong solutions to the equation. Second, we establish two blow-up results. Third, we present the blow-up rate of strong solutions to the equation. Finally, we prove that the equation also has global solutions.