On the parabolic-elliptic Keller-Segel system with signal-dependent motilities: a paradigm for global boundedness and steady states

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 in \cite{KS-1971-JTB2} to describe the aggregation phase of {\it 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.

• Publication year

  2020

• Venue

  arXiv: Analysis of PDEs

• Publication date

  2020-05-09

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.22541/au.159317660.09415314
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Diffusion

  2021cited by this paper
• Stationary and non-stationary patterns of the density-suppressed motility model

  2020cited by this paper
• Comparison methods for a Keller–Segel-type model of pattern formations with density-suppressed motilities

  2020cited by this paper
• Global existence for a kinetic model of pattern formation with density-suppressed motilities

  2020cited by this paper
• Global dynamics and spatio-temporal patterns of predator–prey systems with density-dependent motion

  2019cited by this paper
• Critical mass on the Keller-Segel system with signal-dependent motility

  2019cited by this paper
• Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth

  2019cited by this paper
• Uniqueness and convergence on equilibria of the Keller–Segel system with subcritical mass

  2018cited by this paper
• Pattern formation in bacterial colonies with density-dependent diffusion

  2018cited by this paper
• Boundedness, Stabilization, and Pattern Formation Driven by Density-Suppressed Motility

  2018cited by this paper
• Global Existence and Aggregation in a Keller–Segel Model with Fokker–Planck Diffusion

  2017cited by this paper
• Boundary concentration phenomena for the higher-dimensional Keller–Segel system

  2016cited by this paper
• Boundedness in a fully parabolic chemotaxis system with singular sensitivity

  2015cited by this paper
• Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

  2015cited by this paper
• Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant

  2014cited by this paper
• Competing effects of attraction vs. repulsion in chemotaxis

  2013cited by this paper
• Stripe formation in bacterial systems with density-suppressed motility.

  2012cited by this paper
• Sequential Establishment of Stripe Patterns in an Expanding Cell Population

  2011influential reference
• Transport Equations in Biology

  2006influential reference
• Collapsing steady states of the Keller–Segel system

  2006cited by this paper
• Chemotaxis

  2005cited by this paper
• Steady State Solutions of a Reaction‐Diffusion System Modeling Chemotaxis

  2002cited by this paper
• Blowup of nonradial solutions to parabolic–elliptic systems modeling chemotaxis in two-dimensional domains

  2001cited by this paper
• Chemotactic collapse in a parabolic-elliptic system of mathematical biology

  1999cited by this paper
• DIFFUSION, CROSS-DIFFUSION, AND THEIR SPIKE-LAYER STEADY STATES

  1998cited by this paper
• Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems

  1993cited by this paper
• Locating the peaks of least energy solutions to a semilinear Neumann problem

  1993cited by this paper
• On the shape of least‐energy solutions to a semilinear Neumann problem

  1991cited by this paper
• Large amplitude stationary solutions to a chemotaxis system

  1988influential reference
• Swarms of Predators Exhibit "Preytaxis" if Individual Predators Use Area-Restricted Search

  1987cited by this paper
• On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type

  1986cited by this paper
• LP Bounds of solutions of reaction-diffusion equations

  1979cited by this paper
• Chemotaxis, signal relaying and aggregation morphology.

  1973influential reference
• Semi-linear second-order elliptic equations in L 1

  1973cited by this paper
• Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I

  1959cited by this paper
• A User's Guide to Pde Models for Chemotaxis

  year unknowninfluential reference
• F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences

  year unknowncited by this paper
• ftp ejde.math.txstate.edu (login: ftp) TWO-DIMENSIONAL KELLER-SEGEL MODEL: OPTIMAL CRITICAL MASS AND QUALITATIVE PROPERTIES OF THE SOLUTIONS

  year unknowninfluential reference

• Global dynamics and pattern formation for predator-prey system with density-dependent motion.

  2023cites this paper
• Global solutions to a chemotaxis consumption model involving signal-dependent degenerate diffusion and logistic-type dampening

  2023cites this paper
• Boundedness and stabilization of a predator-prey model with attraction- repulsion taxis in all dimensions.

  2022cites this paper
• Boundedness and exponential stabilization for time-space fractional parabolic-elliptic Keller-Segel model in higher dimensions

  2022cites this paper
• Global Boundedness of the Fully Parabolic Keller-Segel System with Signal-Dependent Motilities

  2021cites this paper
• Global existence, uniform boundedness, and stabilization in a chemotaxis system with density-suppressed motility and nutrient consumption

  2021cites this paper
• On the parabolic‐elliptic Keller–Segel system with signal‐dependent motilities: A paradigm for global boundedness and steady states

  2020cites this paper
• Boundedness of Classical Solutions to a Degenerate Keller–Segel Type Model with Signal-Dependent Motilities

  2020cites this paper
• Global Boundedness of the Fully Parabolic Keller-Segel System with Signal-Dependent Motilities

  2020cites this paper
• Boundedness and Exponential Stabilization in a Parabolic-Elliptic Keller—Segel Model with Signal-Dependent Motilities for Local Sensing Chemotaxis

  2020cites this paper