Blow-up phenomena and global existence for the nonuniform weakly dissipative b-equation

We first establish the local well-posedness for the nonuniform weakly dissipative b-equation which includes both the weakly dissipative Camassa–Holm equation and the weakly dissipative Degasperis–Procesi equation as its special cases. We then study the blow-up phenomena and the long time behavior of the solutions. Two blow-up results are established for certain initial profiles. Moreover, two sufficient conditions for the decay of the solutions are presented.