LAUSR

In this paper, we mainly revisit a weakly dissipative periodic two‐component b$$ b $$ ‐family system. Considering the dissipative effect, the local well‐posedness is first obtained for the system by applying the Kato's semigroup theory. We then utilize the characteristics line method to get one blow‐up criterion with the dispersive parameter k1∈(1,3]$$ {k}_1\in \left(1,3\right] $$ . Finally, the other blow‐up criterion is derived with regard as considering the odevity of initial data by using energy method. When the parameters λ,ki(i=1,2,3)$$ \lambda, {k}_i\left(i=1,2,3\right) $$ belong to suitable range, our obtained results supplement the corresponding blow‐up criteria, which was derived by Liu and Yin (Nonlinear Analysis: Real World Applications 12 (2011) 3608–3620).