Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals

This paper considers the two-species chemotaxis system with two chemicals in a smooth bounded domain , subject to the non-flux boundary condition, and . We obtain a blow-up criterion that if , then there exist finite time blow-up solutions to the system with and . When , the blow-up criterion becomes , and the global boundedness of solutions is furthermore established with under the condition that . This improves the current results for finite time blow-up with and global boundedness with respectively in Tao and Winkler (2015 Discrete Contin. Dyn. Syst. Ser. B 20 3165–83)