Boundedness and Large Time Behavior for Flux Limitation in a Two-Species Chemotaxis System

Abstract

In this paper, the following cross-diffusion system is investigated

$$\begin{aligned} \left\{ \begin{array}{llll} u_t=\nabla \cdot \big ((u+1)^{m-1}\nabla u\big )-\nabla \cdot \Bigg (\frac{u\nabla z}{(1+|\nabla z|^2)^\alpha }\Bigg ),\\ 0=\Delta z-z+v,\\ v_t=\nabla \cdot \big ((v+1)^{m-1}\nabla v\big )-\nabla \cdot \Bigg (\frac{v\nabla w}{(1+|\nabla w|^2)^\alpha }\Bigg ),\\ 0=\Delta w-w+u, \end{array}\right. \end{aligned}$$

in a bounded domain $\Omega \subset {\mathbb {R}}^n$ ($n\ge 2$) with smooth boundary $\partial \Omega$. Under the condition that $\alpha >\frac{2n-mn-2}{2(n-1)}$ and $m\ge 1,$ it is shown that the problem possesses a global bounded classical solution. Moreover, we also investigated the large time behavior of the solution, and obtained the corresponding solution exponentially converges to a constant stationary solution when the initial data $u_0$ is sufficiently small.