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) = del center dot (u del u/root u(2) + |del u|(2)) - chi del center dot (uf(|del v|(2))del v); x is an element of Omega; t > 0, 0 = del v - mu + u; x is an element of Omega; t > 0; (star) under no-flux boundary conditions in a ball Omega = B-R(0) subset of R-n (n >= 1), where mu := 1/|Omega| integral(Omega) u(0)(x) dx is positive and f(xi) = (1 + xi)(-alpha), for xi >= 0 with alpha > 0. It is proved that the problem (star) possesses a unique classical solution that can be extended in time up to a maximal T-max is an element of(0, infinity]. Moreover, it is shown that the above solution is global and bounded when either chi < (1+m(2))(alpha)/m if n = 1 and alpha <= 1/2, or chi< root 2 alpha - 1 (2 alpha/2 alpha-1)(alpha) if n >= 1 and alpha > 1/2. We point out that when alpha = 1/2, our result is consistent with that of [N. Bellomo and M. Winkler, Comm. Partial Differential Equations 42 (2017) 436-473].