Global Bounded Solutions and Large Time Behavior of a Chemotaxis System with Flux Limitation

In this paper, the following cross-diffusion system is investigated {u(t)=del<middle dot>((u+1)(m)del u)-del<middle dot>(u(u+1)(beta-1)del v/(1+|del v|(2))(alpha))+a-bu (R),x is an element of Omega, t >0, 0=Delta v-v+u,x is an element of Omega, t >0, in a bounded domain Omega subset of Rn(n >= 2) with smooth boundary partial derivative Omega. Under the condition that alpha>2n-mn-2/2(n-1),m >= 1, and beta <= m+2/2, it is shown that the problem possesses a unique global bounded classical solution. Moreover, it is obtained that the corresponding solution exponentially converge to a constant stationary solution when the initial data u(0) is sufficiently small.