Collision avoidance and asymptotic flocking in the delayed Cucker−Smale model with singular short range communication weights

The delayed Cucker−Smale (C−S) flocking model has been widely studied, however, only when the communication weights are regular. For the singular short range communication case, collision avoidance is the key of both well-posedness and large time behavior. In this paper, by an inequality of second-order velocity-position moment, collision avoidance1 is proved for the delayed C−S with some singular short range communication weights. Then, asymptotic flocking is established for some restricted initial data.