Global boundedness and large time behavior of solutions to a chemotaxis system with flux limitation

Abstract

This paper deals with the following cross-diffusion system{ut=∇⋅((u+1)m−1∇u)−∇⋅(u∇v(1+|∇v|2)α),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0, in a bounded domain Ω⊂Rn, n≥2 with smooth boundary. In this framework, it is shown that the problem possesses a unique global bounded classical solution under the condition that α>2n−2−mn2(n−1) and m≥1. Moreover, it is asserted that the corresponding solution exponentially converges to the constant stationary solution (u0‾,u0‾) when the initial data u0 is sufficiently small, where u0‾:=∫Ωu0|Ω|.