## Abstract

In the paper we first establish the local well-posedness for a family of nonlinear dispersive equations, the so called *b*-equation. Then we describe the precise blow-up scenario. Moreover, we prove that for the *b*-equation we do have the coexistence of global in time solutions and blow-up phenomena: Depending on the initial data solutions may exist for ever, while other data force the solution to produce a singularity in finite time. Finally, we prove the uniqueness and existence of global weak solution to the equation provided the initial data satisfy certain sign conditions.

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