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<p>In this paper, we investigate a chemotaxis system with signal-dependent motility and indirect signal production/consumption in bounded domains (dimensions 1-5). For two fundamental cases, signal consumption (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f=-vw\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mo>-</mo> <mi>v</mi> <mi>w</mi> </mrow> </math></EquationSource> </InlineEquation>) with general positive motility, and signal production (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(f=-v+w\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mo>-</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> </mrow> </math></EquationSource> </InlineEquation>) with bounded, differentiable motility, we establish the existence of global classical solutions that remain uniformly bounded in time. These results extend previous works to broader conditions and higher dimensions. Moreover, using Lyapunov functionals, we characterize the long-term dynamics: Solutions converge exponentially to homogeneous steady states, with the equilibrium pattern differing between consumption (semitrivial) and production (fully populated) cases. When the signal diffuses sufficiently fast in the production scenario, all components stabilize to the mass-averaged initial cell density. The analysis develops new techniques to overcome the challenges posed by the nonlinear coupling and higher-dimensional settings.</p>

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Global dynamics in chemotaxis systems with signal-dependent motility and indirect signal mechanisms

  • 期刊论文

  • 发表日期：

    2025年12月26日

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  • Changfeng Liu,
  • Jianping Gao,
  • Shangjiang Guo

摘要

In this paper, we investigate a chemotaxis system with signal-dependent motility and indirect signal production/consumption in bounded domains (dimensions 1-5). For two fundamental cases, signal consumption ( \(f=-vw\) f = - v w ) with general positive motility, and signal production ( \(f=-v+w\) f = - v + w ) with bounded, differentiable motility, we establish the existence of global classical solutions that remain uniformly bounded in time. These results extend previous works to broader conditions and higher dimensions. Moreover, using Lyapunov functionals, we characterize the long-term dynamics: Solutions converge exponentially to homogeneous steady states, with the equilibrium pattern differing between consumption (semitrivial) and production (fully populated) cases. When the signal diffuses sufficiently fast in the production scenario, all components stabilize to the mass-averaged initial cell density. The analysis develops new techniques to overcome the challenges posed by the nonlinear coupling and higher-dimensional settings.