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 u t = ∇ ⋅ u ∇ u u 2 + | ∇ u | 2 − χ ∇ ⋅ (u f (| ∇ v | 2) ∇ v) , x ∈ Ω ,   t > 0 , 0 = Δ v − μ + u , x ∈ Ω ,   t > 0 , (⋆) under no-flux boundary conditions in a ball Ω = B R (0) ⊂ ℝ n (n ≥ 1), where μ : = 1 | Ω | ∫ Ω u 0 (x) d x is positive and f (ξ) = (1 + ξ) − α , for ξ ≥ 0 with α > 0. It is proved that the problem (⋆) possesses a unique classical solution that can be extended in time up to a maximal T max ∈ (0 , ∞ ]. Moreover, it is shown that the above solution is global and bounded when either χ < (1 + m 2) α m if n = 1 and α ≤ 1 2 , or χ < 2 α − 1 (2 α 2 α − 1) α if n ≥ 1 and α > 1 2 . We point out that when α = 1 2 , our result is consistent with that of [N. Bellomo and M. Winkler, Comm. Partial Differential Equations 42 (2017) 436–473]. [ABSTRACT FROM AUTHOR]