This paper deals with the chemotaxis consumption model with signal-dependent motility

under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset {\mathbb {R}}^n\) (\(n\ge 2\)), the initial data \(u_0\in C^0({\overline{\Omega }})\) and \(v_0\in W^{1,\infty }(\Omega )\) are non-negative and the parameter \(\mu >0\). The motility function r(v) satisfies \(r(v)\in C^3([0,\infty ))\) with \(r(v)>0\) and \(r'(v)<0\) for all \(v\ge 0\). When \(n=2\) and \(\mu >0\), we proved that the system admits a globally bounded classical solution. When \(n\ge 3\), we establish the global existence and the boundedness of the solution if \(\mu \) is suitably large. Moreover, by constructing Lyapunov functions it is shown that the global bounded classical solution will exponentially converge to (1, 0) as \(t\rightarrow \infty \).

本文讨论了具有信号依赖性运动的趋化性消耗模型

在光滑有界域\(\Omega \subset {\mathbb {R}}^n\) ( \(n\ge 2\) ) 中的齐次 Neumann 边界条件下，初始数据\(u_0\in C^0( {\overline{\Omega }})\)和\(v_0\in W^{1,\infty }(\Omega )\)是非负的，参数\(\mu >0\)。运动函数r ( v ) 满足\(r(v)\in C^3([0,\infty ))\)与\(r(v)>0\)和\(r'(v)<0 \)对于所有\(v\ge 0\)。当\(n=2\)和\(\mu >0\) 时，我们证明了系统承认全局有界经典解。当\(n\ge 3\)，如果\(\mu \)适当大，我们建立解决方案的全局存在性和有界性。此外，通过构造李雅普诺夫函数，表明全局有界经典解将指数收敛到 (1, 0) 为\(t\rightarrow \infty \)。