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.