Local existence and global boundedness for a chemotaxis system with gradient dependent flux limitation

In this paper, we investigate the following Keller–Segel system with flux limitation Formula: see text under no-flux boundary conditions in a ball Formula: see text), where Formula: see text is positive and Formula: see text with Formula: see text. It is proved that the problem (⋆) possesses a unique classical solution that can be extended in time up to a maximal Formula: see text. Moreover, it is shown that the above solution is global and bounded when either Formula: see text if Formula: see text and Formula: see text, or Formula: see text if Formula: see text and Formula: see text. We point out that when Formula: see text, our result is consistent with that of N. Bellomo and M. Winkler, Comm. Partial Differential Equations 42 (2017) 436–473.