Global boundedness in a Keller-Segel system with flux limitation and logistic source

We consider the parabolic chemotaxis system ut = & UDelta;u - V & BULL; (uf (|Vv|2)Vv) + ru - & mu;u & gamma;, x E & omega;, t > 0, vt = & UDelta;v - v + u, x E & omega;, t > 0, in a smooth bounded domain & omega; C Rn(n > 1) with the homogeneous Neumann boundary conditions, where r, & mu; > 0, & gamma; > 1 and the function f satisfies f(& xi;) = (1 + & xi;)- & alpha;2 , for all & xi; > 0, with & alpha; > 0. In the case n < 2, it is shown that the corresponding initial value problem possesses a global bounded classical solution for any & alpha;, & mu; > 0. In the case n > 3, if & gamma; = 2 and & alpha; = n-2 2n , there exists & mu;0 > 0 such that for any & mu; > & mu;0, a global bounded classical solution exists.