This paper is concerned with a parabolic-elliptic Keller–Segel system where both diffusive and chemotactic coefficients (motility functions) depend on the chemical signal density. This system was originally proposed by Keller and Segel to describe the aggregation phase of Dictyostelium discoideum cells in response to the secreted chemical signal cyclic adenosine monophosphate (cAMP), but the available analytical results are very limited by far. Considering system in a bounded smooth domain with Neumann boundary conditions, we establish the global boundedness of solutions in any dimensions with suitable general conditions on the signal-dependent motility functions, which are applicable to a wide class of motility functions. The existence/nonexistence of non-constant steady states is studied and abundant stationary profiles are found. Some open questions are outlined for further pursues. Our results demonstrate that the global boundedness and profile of stationary solutions to the Keller–Segel system with signal-dependent motilities depend on the decay rates of motility functions, space dimensions and the relation between the diffusive and chemotactic motilities, which makes the dynamics immensely wealthy.

中文翻译：

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关于具有信号依赖运动的抛物线-椭圆 Keller-Segel 系统：全局有界和稳态的范式

本文涉及抛物线-椭圆 Keller-Segel 系统，其中扩散系数和趋化系数（运动函数）都取决于化学信号密度。该系统最初由 Keller 和 Segel 提出，用于描述盘基网柄菌的聚集阶段细胞对分泌的化学信号环磷酸腺苷 (cAMP) 做出反应，但目前可用的分析结果非常有限。考虑在具有 Neumann 边界条件的有界光滑域中的系统，我们建立了任何维度的解的全局有界性，并在信号相关的运动函数上建立了合适的一般条件，适用于广泛的运动函数类。研究了非恒定稳态的存在/不存在，并发现了丰富的平稳分布。概述了一些开放性问题以供进一步研究。我们的结果表明，具有信号依赖性运动的 Keller-Segel 系统的平稳解的全局有界性和轮廓取决于运动函数的衰减率，