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

In this paper, the following cross-diffusion system is investigated \(\begin{aligned} \left\{ \begin{array}{llll} u_t=\nabla \cdot \big ((u+1)^{m-1}\nabla u\big )-\nabla \cdot \Bigg (\frac{u\nabla z}{(1+|\nabla z|^2)^\alpha }\Bigg ),\\ 0=\Delta z-z+v,\\ v_t=\nabla \cdot \big ((v+1)^{m-1}\nabla v\big )-\nabla \cdot \Bigg (\frac{v\nabla w}{(1+|\nabla w|^2)^\alpha }\Bigg ),\\ 0=\Delta w-w+u, \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t=\mathrm{\nabla }·({(u+1)}^{m-1}\mathrm{\nabla }u)-\mathrm{\nabla }·(\frac{u\mathrm{\nabla }z}{{(1+|\mathrm{\nabla }z|}^2{)}^{\alpha }}),\hfill \\ 0=\mathrm{\Delta }z-z+v,\hfill \\ v_t=\mathrm{\nabla }·({(v+1)}^{m-1}\mathrm{\nabla }v)-\mathrm{\nabla }·(\frac{v\mathrm{\nabla }w}{{(1+|\mathrm{\nabla }w|}^2{)}^{\alpha }}),\hfill \\ 0=\mathrm{\Delta }w-w+u,\hfill \end{array}\right)\end{array}$$ in a bounded domain \(\Omega \subset {\mathbb {R}}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ ( \(n\ge 2\) $n\ge 2$ ) with smooth boundary \(\partial \Omega \) $\partial \mathrm{\Omega }$ . Under the condition that \(\alpha >\frac{2n-mn-2}{2(n-1)}\) $\alpha >\frac{2n-mn-2}{2(n-1)}$ and \(m\ge 1,\) $m\ge 1,$ 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 \(u_0\) $u_0$ is sufficiently small.