Exponential convergence to equilibrium in a coupled gradient flow system modeling chemotaxis

We study a system of two coupled nonlinear parabolic equations. It constitutes a variant of the Keller-Segel model for chemotaxis, i.e. it models the behaviour of a population of bacteria that interact by means of a signalling substance. We assume an external confinement for the bacteria and a nonlinear dependency of the chemotactic drift on the signalling substance concentration. We perform an analysis of existence and long-time behaviour of solutions based on the underlying gradient flow structure of the system. The result is that, for a wide class of initial conditions, weak solutions exist globally in time and converge exponentially fast to the unique stationary state under suitable assumptions on the convexity of the confinement and the strength of the coupling.

[1]  Yoshie Sugiyama,et al.  Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term , 2006 .

[2]  C. Villani.  Topics in Optimal Transportation , 2003 .

[3]  James D. Murray,et al.  Spatial models and biomedical applications , 2003 .

[4]  Bogdan-Vasile Matioc,et al.  A gradient flow approach to a thin film approximation of the Muskat problem , 2013 .

[5]  Adrien Blanchet,et al.  A GRADIENT FLOW APPROACH TO THE KELLER-SEGEL SYSTEMS (Progress in Variational Problems : Variational Problems Interacting with Probability Theories) , 2013 .

[6]  T. Yamada.  Improvement of convergence rates for a parabolic system of chemotaxis in the whole space , 2011 .

[7]  Eric Carlen,et al.  Functional inequalities, thick tails and asymptotics for the critical mass Patlak–Keller–Segel model , 2010, 1009.0134.

[8]  Martial Agueh,et al.  Rates of decay to equilibria for p-Laplacian type equations , 2008 .

[9]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[10]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[11]  The Basin of Attraction of the Steady-States for a Chemotaxis Model in $${\mathbb{R}^2}$$ with Critical Mass , 2013 .

[12]  D. Kinderlehrer,et al.  THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION , 1996 .

[13]  S. Luckhaus,et al.  Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases , 2007 .

[14]  Philippe Laurençot,et al.  The Parabolic-Parabolic Keller-Segel System with Critical Diffusion as a Gradient Flow in ℝ d , d ≥ 3 , 2012, 1203.3573.

[15]  Lorenzo Giacomelli,et al.  Variatonal formulation for the lubrication approximation of the Hele-Shaw flow , 2001 .

[16]  T. Senba,et al.  A quasi-linear parabolic system of chemotaxis , 2006 .

[17]  Katharina Post,et al.  A Non-linear Parabolic System Modeling Chemotaxis with Sensitivity Functions , 1999 .

[18]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[19]  H. Kozono,et al.  Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces , 2009 .

[20]  José A. Carrillo,et al.  Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak-Keller-Segel Model , 2008, SIAM J. Numer. Anal..

[21]  J. A. Carrillo,et al.  Asymptotic L1-decay of solutions of the porous medium equation to self-similarity , 2000 .

[22]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[23]  C. Villani,et al.  Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media , 2006 .

[24]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[25]  Yoshie Sugiyama,et al.  Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems , 2006 .

[26]  B. Perthame.  Transport Equations in Biology , 2006 .

[27]  Vincent Calvez,et al.  Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities , 2010, 1007.2837.

[28]  N. Mizoguchi.  Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane , 2013 .

[29]  Benoit Perthame,et al.  Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces , 2008, Math. Comput. Model..

[30]  Vincent Calvez,et al.  The parabolic-parabolic Keller-Segel model in R2 , 2008 .

[31]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[32]  R. McCann,et al.  A Family of Nonlinear Fourth Order Equations of Gradient Flow Type , 2009, 0901.0540.

[33]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[34]  K. Painter,et al.  A User's Guide to Pde Models for Chemotaxis , 2022 .

[35]  Y. Sugiyama.  Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis , 2007, Differential and Integral Equations.

[36]  J. V. Hurley,et al.  Chemotaxis , 2005, Infection.

[37]  G. Burton.  TOPICS IN OPTIMAL TRANSPORTATION (Graduate Studies in Mathematics 58) By CÉDRIC VILLANI: 370 pp., US$59.00, ISBN 0-8218-3312-X (American Mathematical Society, Providence, RI, 2003) , 2004 .

[38]  Michael Winkler,et al.  Absence of collapse in a parabolic chemotaxis system with signal‐dependent sensitivity , 2010 .

[39]  Marco Di Francesco,et al.  Fully parabolic Keller–Segel model for chemotaxis with prevention of overcrowding , 2008 .

[40]  T. Nagai,et al.  Decay Properties and Asymptotic Profiles of Bounded Solutions to a Parabolic System of Chemotaxis in Rn , 2003 .

[41]  T. Zhukovskaya,et al.  Functional Inequalities , 2021, Inequalities in Analysis and Probability.

[42]  E. Lieb,et al.  Analysis, Second edition , 2001 .

[43]  Lee A. Segel,et al.  A Theoretical Study of Receptor Mechanisms in Bacterial Chemotaxis , 1977 .

[44]  Dirk Horstmann,et al.  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 , 2022 .

[45]  E. Zeidler.  Nonlinear Functional Analysis and Its Applications: II/ A: Linear Monotone Operators , 1989 .

[46]  J. Carrillo,et al.  Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions , 2008, 0801.2310.

[47]  Marion Kee,et al.  Analysis , 2004, Machine Translation.

[48]  Y. Sugiyama.  Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems , 2006, Differential and Integral Equations.

[49]  S. Mischler,et al.  Uniqueness and long time asymptotics for the parabolic–parabolic Keller–Segel equation , 2014, 1406.6006.

[50]  Piotr Biler,et al.  Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis , 2009, Journal of mathematical biology.

[51]  Giuseppe Savaré,et al.  The Wasserstein Gradient Flow of the Fisher Information and the Quantum Drift-diffusion Equation , 2009 .

[52]  I. Lapidus,et al.  Model for the chemotactic response of a bacterial population. , 1976, Biophysical journal.

[53]  J. Carrillo,et al.  Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations , 2011 .

[54]  D. Kinderlehrer,et al.  A HYBRID VARIATIONAL PRINCIPLE FOR THE KELLER-SEGEL SYSTEM IN R 2 , 2015 .

[55]  Eberhard Zeidler,et al.  Nonlinear monotone operators , 1990 .

[56]  Marco Di Francesco,et al.  Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for ut = Δϕ(u) , 2006 .

[57]  Jonathan Zinsl.  Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure , 2014 .

[58]  C. Villani,et al.  Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates , 2003 .