Critical Finite‐Time Blow‐Up Line in a Two‐Species Quasilinear Chemotaxis System With Two Chemicals

In this study, we explore the quasilinear two‐species chemotaxis system with two chemicals where () is a smooth bounded domain. The functions and exhibit asymptotic behavior of the form We prove that when is a ball, if and , there exist radially symmetric initial data and , such that the corresponding solutions blow up in finite time; for any general smooth bounded domain , if , all solutions are global. The result in [X. Huang and H. Zhong, Blow‐up curve for a quasilinear two‐species chemotaxis system with two chemicals in higher dimension, Mathematical Methods in the Applied Sciences, 2026] implies that all solutions are globally bounded when Combining these results, we point out that the system () possesses two critical lines and to classify three dynamics among global boundedness, finite‐time blow‐up, and global existence of solutions to system ().