Global solutions and blow‐up phenomena for the periodic b ‐equation

In the paper, we mainly study the Cauchy problem of a family of asymptotically equivalent shallow water wave equations, the so called periodic b ‐equation. We first establish the local well‐posedness for the periodic b ‐equation. Then we derive the precise blow‐up scenario and present two blow‐up results. Moreover, we show that the periodic b ‐equation has global strong solutions. Finally, we prove the existence of global weak solutions to the periodic b ‐equation, provided that initial data satisfy certain sign conditions.