Blow-up and boundedness in a chemotaxis system with flux limitation and nonlinear diffusion

This paper deals with the following chemotaxis system with gradient-dependent flux limitation and nonlinear diffusion Formula: see text under homogeneous Neumann boundary conditions in a smoothly bounded domain Formula: see text Formula: see text, where Formula: see text, Formula: see text, Formula: see text generalizes the prototype given by Formula: see text for all Formula: see text with Formula: see text, Formula: see text. It is shown that the corresponding initial-boundary value problem possesses a unique globally bounded classical solution provided that Formula: see text and Formula: see text. While for Formula: see text and Formula: see text, there exist nonnegative radially symmetric initial data such that the solution blows up in finite time. These conditions are optimal for Formula: see text, especially when Formula: see text, which is consistent with the results in Winkler Indiana Univ. Math. J. 71 (2022) 1437–1465. Furthermore, along the critical line Formula: see text with Formula: see text, the finite-time blow-up of radial solutions with sufficiently large initial mass is also established, conversely, the global boundedness of solutions with suitably small initial data is proved.