Boundedness and Finite-Time Blow-up in a Chemotaxis System with Flux Limitation and Logistic Source

The chemotaxis system ut = u-.. center dot (u|. v|p-2. v)+.u- mu u., 0 = v + u- h(u, v) (*) is considered in a smoothly bounded domain . Rn (n. N), where. > 0, p > 1,. = 0, mu > 0,. > 1, and h = v or h = 1 | | u. It is firstly proved that if n = 1 and p > 1 is arbitrary, or n = 2 and p. (1, n n-1), then for all continuous initial data a corresponding noflux type initial-boundary value problem for (*) admits a globally defined and bounded weak solution. Secondly, it is shown that if n = 2, = BR(0). Rn is a ball with some R > 0, p > n n-1 and. > 1 is small enough, then one can find a nonnegative radially symmetric function u0 and a weak solution of (*) with initial datum u0 which blows up in finite time.