Asymptotic behavior of solutions to a singular chemotaxis system in multi-dimensions

In this paper, we investigate the optimal large-time behavior of the global solution to a singular chemotaxis system in the whole space $\mathbb{R}^d$ with $d=2,3$. Assuming that the initial data is sufficiently close to an equilibrium state, we first prove the $k$-th order spatial derivative of the global solution converges to its corresponding equilibrium at the optimal rate $(1+t)^{-(\frac{d}{4}+\frac{k}{2})}$, which improve upon the result in [37]. Then, for well-chosen initial data, we also establish lower bounds on the convergence rates, which match those of the heat equation. Our proof relies on a Cole-Hopf type transformation, delicate spectral analysis, the Fourier splitting technique, and energy methods.

• Publication year

  2025

• Venue

  Unknown venue

• Publication date

  2025-12-02

• Fields of study

  Mathematics

• Identifiers
  arXiv 2512.02583
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Cauchy problem of a system of parabolic conservation laws arising from the singular Keller-Segel model in multi-dimensions

  2021influential reference
• THE SHARP TIME DECAY RATE OF THE ISENTROPIC NAVIER-STOKES SYSTEM IN R 3

  2020cited by this paper
• Optimal decay rates for a chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate

  2020cited by this paper
• The large-time behavior of the multi-dimensional hyperbolic-parabolic model arising from chemotaxis

  2019cited by this paper
• Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions

  2019influential reference
• Contraction for large perturbations of traveling waves in a hyperbolic–parabolic system arising from a chemotaxis model

  2019cited by this paper
• Boundary layers and stabilization of the singular Keller-Segel system

  2018cited by this paper
• Nonlinear stability of strong traveling waves for the singular Keller–Segel system with large perturbations

  2018cited by this paper
• Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption

  2018cited by this paper
• Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology

  2018cited by this paper
• Stability of Boundary Layers for a Viscous Hyperbolic System Arising from Chemotaxis: One-Dimensional Case

  2018cited by this paper
• The Optimal Convergence Rates for the Multi-dimensional Chemotaxis Model in Critical Besov Spaces

  2016cited by this paper
• Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis

  2016influential reference
• The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties

  2016cited by this paper
• Boundary layer problem on a hyperbolic system arising from chemotaxis

  2016cited by this paper
• Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

  2015cited by this paper
• Initial–boundary value problems for a system of hyperbolic balance laws arising from chemotaxis

  2015cited by this paper
• Quantitative decay of a one-dimensional hybrid chemotaxis model with large data

  2015influential reference
• Stability of traveling waves of the Keller–Segel system with logarithmic sensitivity

  2014cited by this paper
• Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model

  2013cited by this paper
• Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity

  2013cited by this paper
• Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis

  2012influential reference
• LARGE-TIME BEHAVIOR OF A PARABOLIC-PARABOLIC CHEMOTAXIS MODEL WITH LOGARITHMIC SENSITIVITY IN ONE DIMENSION

  2012cited by this paper
• Blow up criterion for a hyperbolic–parabolic system arising from chemotaxis

  2012cited by this paper
• Steadily propagating waves of a chemotaxis model.

  2012cited by this paper
• Global Dynamics of a Hyperbolic-Parabolic Model Arising from Chemotaxis

  2012cited by this paper
• Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework

  2012influential reference
• Global Large-Data Solutions in a Chemotaxis-(Navier–)Stokes System Modeling Cellular Swimming in Fluid Drops

  2012cited by this paper
• ON A HYPERBOLIC–PARABOLIC SYSTEM MODELING CHEMOTAXIS

  2011cited by this paper
• Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis

  2011cited by this paper
• Logarithmic sensing in Escherichia coli bacterial chemotaxis.

  2009cited by this paper
• Shock formation in a chemotaxis model

  2008influential reference
• Mathematical Analysis of a Model for the Initiation of Angiogenesis

  2002cited by this paper
• Vorticity and incompressible flow

  2001cited by this paper
• A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. I. The role of protease inhibitors in preventing angiogenesis.

  2000cited by this paper
• Aggregation, Blowup, and Collapse: The ABC's of Taxis in Reinforced Random Walks

  1997cited by this paper
• A System of Reaction Diffusion Equations Arising in the Theory of Reinforced Random Walks

  1997influential reference
• Traveling bands of chemotactic bacteria: a theoretical analysis.

  1971cited by this paper
• Model for chemotaxis.

  1971cited by this paper
• Chemotaxis in Bacteria

  1966cited by this paper
• A User's Guide to Pde Models for Chemotaxis

  year unknowncited by this paper

• No citing papers are available for this paper.