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

Abstract

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

$$ \textstyle\begin{cases} u_{t}=\nabla \cdot \big((u+1)^{m}\nabla u\big)-\nabla \cdot \Bigg( \frac{u(u+1)^{\beta -1}\nabla v}{(1+|\nabla v|^{2})^{\alpha }}\Bigg)+a-bu^{r}, \,\,& x\in \Omega ,\,\,t>0, \\ 0=\Delta v-v+u, & x\in \Omega ,\,\,t>0, \end{cases} $$

in a bounded domain \(\Omega \subset \mathbb{R}^{n}\) (\(n\ge 2\)) with smooth boundary \(\partial \Omega \). Under the condition that \(\alpha >\frac{2n-mn-2}{2(n-1)}\), \(m\geq 1\), and \(\beta \leq \frac{m+2}{2}\), 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.