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April  2021, 26(4): 1931-1966. doi: 10.3934/dcdsb.2020326

## Existence and uniqueness of solutions for a hyperbolic Keller–Segel equation

 Université de Bordeaux – Institut de Mathématiques de Bordeaux, 351 cours de la Libération, 33400 Talence, France

* Corresponding author: Pierre Magal, pierre.magal@u-bordeaux.fr

Received  August 2020 Published  April 2021 Early access  November 2020

Fund Project: The research of the first author is supported by China Scholarship Council

In this work we describe a hyperbolic model with cell-cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call "pressure") which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We analyze the well-posed property of the associated Cauchy problem on the real line. Moreover we obtain a convergence result for bounded initial distributions which are positive and stay away from zero uniformly on the real line.

Citation: Xiaoming Fu, Quentin Griette, Pierre Magal. Existence and uniqueness of solutions for a hyperbolic Keller–Segel equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1931-1966. doi: 10.3934/dcdsb.2020326
##### References:
 [1] N. J. Armstrong, K. J. Painter and J. A. Sherratt, A continuum approach to modelling cell-cell adhesion, J. Theoret. Biol., 243 (2006), 98-113.  doi: 10.1016/j.jtbi.2006.05.030. [2] D. G. Aronson, Density-dependent interaction-diffusion systems, Dynamics and Modelling of Reactive Systems, Publ. Math. Res. Center Univ. Wisconsin, Academic Press, New York-London, 44 (1980), 161-176. [3] C. Atkinson, G. E. H. Reuter and C. J. Ridler-Rowe, Traveling wave solution for some nonlinear diffusion equations, SIAM J. Math. Anal., 12 (1981), 880-892.  doi: 10.1137/0512074. [4] D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul, Nonlocal interactions by repulsive-attractive potentials: Radial ins/stability, Phys. D, 260 (2013), 5-25.  doi: 10.1016/j.physd.2012.10.002. [5] N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives, Math. Models Methods Appl. Sci., 22 (2012), 1130001, 37 pp. doi: 10.1142/S0218202512005885. [6] A. J. Bernoff and C. M. Topaz, Nonlocal aggregation models: A primer of swarm equilibria [reprint of mr2788924], SIAM Rev., 55 (2013), 709-747.  doi: 10.1137/130925669. [7] A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83.  doi: 10.1002/cpa.20334. [8] M. Burger, R. Fetecau and Y. Huang, Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13 (2014), 397-424.  doi: 10.1137/130923786. [9] V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\Bbb R^2$, Commun. Math. Sci., 6 (2008), 417–447, http://projecteuclid.org/euclid.cms/1214949930. doi: 10.4310/CMS.2008.v6.n2.a8. [10] K. Carrapatoso and S. Mischler, Uniqueness and long time asymptotics for the parabolic-parabolic Keller-Segel equation, Comm. Partial Differential Equations, 42 (2017), 291-345.  doi: 10.1080/03605302.2017.1280682. [11] J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Confinement in nonlocal interaction equations, Nonlinear Anal., 75 (2012), 550-558.  doi: 10.1016/j.na.2011.08.057. [12] J. A. Carrillo, H. Murakawa, M. Sato, H. Togashi and O. Trush, A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation, J. Theoret. Biol., 474 (2019), 14-24.  doi: 10.1016/j.jtbi.2019.04.023. [13] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998. [14] S. Childress, Chemotactic collapse in two dimensions, Modelling of Patterns in Space and Time, Lecture Notes in Biomath., Springer, Berlin, 55 (1984), 61-66.  doi: 10.1007/978-3-642-45589-6_6. [15] A. de Pablo and J. L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equations, 93 (1991), 19-61.  doi: 10.1016/0022-0396(91)90021-Z. [16] A. Ducrot, X. Fu and P. Magal, Turing and Turing-Hopf bifurcations for a reaction diffusion equation with nonlocal advection, J. Nonlinear Sci., 28 (2018), 1959-1997.  doi: 10.1007/s00332-018-9472-z. [17] A. Ducrot and P. Magal, Asymptotic behavior of a nonlocal diffusive logistic equation, [18] A. Ducrot and D. Manceau, A one-dimensional logistic like equation with nonlinear and nonlocal diffusion: Strong convergence to equilibrium, Proc. Amer. Math. Soc., 148 (2020), 3381-3392.  doi: 10.1090/proc/14971. [19] J. Dyson, S. A. Gourley, R. Villella-Bressan and G. F. Webb, Existence and asymptotic properties of solutions of a nonlocal evolution equation modeling cell-cell adhesion, SIAM J. Math. Anal., 42 (2010), 1784-1804.  doi: 10.1137/090765663. [20] R. Eftimie, G. de Vries, M. A. Lewis and F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol., 69 (2007), 1537-1565.  doi: 10.1007/s11538-006-9175-8. [21] X. Fu, Q. Griette and P. Magal, A cell-cell repulsion model on a hyperbolic Keller-Segel equation, J. Math. Biol., 80 (2020), 2257-2300.  doi: 10.1007/s00285-020-01495-w. [22] F. Hamel and C. Henderson, Propagation in a Fisher-KPP equation with non-local advection, J. Funct. Anal., 278 (2020), 108426, 53 pp. doi: 10.1016/j.jfa.2019.108426. [23] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3. [24] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6. [25] S. Katsunuma, H. Honda, T. Shinoda, Y. Ishimoto, T. Miyata, H. Kiyonari, T. Abe, K.-i. Nibu, Y. Takai and H. Togashi, Synergistic action of nectins and cadherins generates the mosaic cellular pattern of the olfactory epithelium, Journal of Cell Biology, 212 (2016), 561-575.  doi: 10.1083/jcb.201509020. [26] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [27] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6. [28] A. J. Leverentz, C. M. Topaz and A. J. Bernoff, Asymptotic dynamics of attractive-repulsive swarms, SIAM J. Appl. Dyn. Syst., 8 (2009), 880-908.  doi: 10.1137/090749037. [29] A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158. [30] A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation, J. Math. Biol., 47 (2003), 353-389.  doi: 10.1007/s00285-003-0209-7. [31] D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66.  doi: 10.1007/s00285-004-0279-1. [32] H. Murakawa and H. Togashi, Continuous models for cell–cell adhesion, Journal of Theoretical Biology, 374 (2015), 1-12.  doi: 10.1016/j.jtbi.2015.03.002. [33] J. Pasquier, P. Magal, C. Boulangé-Lecomte, G. Webb and F. Le Foll, Consequences of cell-to-cell p-glycoprotein transfer on acquired multidrug resistance in breast cancer: a cell population dynamics model, Biol. Direct., 6 (2011), 5. doi: 10.1186/1745-6150-6-5. [34] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407. [35] B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335.  doi: 10.1090/S0002-9947-08-04656-4. [36] W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. [37] Y. Song, S. Wu and H. Wang, Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect, J. Differential Equations, 267 (2019), 6316-6351.  doi: 10.1016/j.jde.2019.06.025. [38] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. [39] E. Zeidler, Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.

show all references

##### References:
 [1] N. J. Armstrong, K. J. Painter and J. A. Sherratt, A continuum approach to modelling cell-cell adhesion, J. Theoret. Biol., 243 (2006), 98-113.  doi: 10.1016/j.jtbi.2006.05.030. [2] D. G. Aronson, Density-dependent interaction-diffusion systems, Dynamics and Modelling of Reactive Systems, Publ. Math. Res. Center Univ. Wisconsin, Academic Press, New York-London, 44 (1980), 161-176. [3] C. Atkinson, G. E. H. Reuter and C. J. Ridler-Rowe, Traveling wave solution for some nonlinear diffusion equations, SIAM J. Math. Anal., 12 (1981), 880-892.  doi: 10.1137/0512074. [4] D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul, Nonlocal interactions by repulsive-attractive potentials: Radial ins/stability, Phys. D, 260 (2013), 5-25.  doi: 10.1016/j.physd.2012.10.002. [5] N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives, Math. Models Methods Appl. Sci., 22 (2012), 1130001, 37 pp. doi: 10.1142/S0218202512005885. [6] A. J. Bernoff and C. M. Topaz, Nonlocal aggregation models: A primer of swarm equilibria [reprint of mr2788924], SIAM Rev., 55 (2013), 709-747.  doi: 10.1137/130925669. [7] A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83.  doi: 10.1002/cpa.20334. [8] M. Burger, R. Fetecau and Y. Huang, Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13 (2014), 397-424.  doi: 10.1137/130923786. [9] V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\Bbb R^2$, Commun. Math. Sci., 6 (2008), 417–447, http://projecteuclid.org/euclid.cms/1214949930. doi: 10.4310/CMS.2008.v6.n2.a8. [10] K. Carrapatoso and S. Mischler, Uniqueness and long time asymptotics for the parabolic-parabolic Keller-Segel equation, Comm. Partial Differential Equations, 42 (2017), 291-345.  doi: 10.1080/03605302.2017.1280682. [11] J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Confinement in nonlocal interaction equations, Nonlinear Anal., 75 (2012), 550-558.  doi: 10.1016/j.na.2011.08.057. [12] J. A. Carrillo, H. Murakawa, M. Sato, H. Togashi and O. Trush, A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation, J. Theoret. Biol., 474 (2019), 14-24.  doi: 10.1016/j.jtbi.2019.04.023. [13] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998. [14] S. Childress, Chemotactic collapse in two dimensions, Modelling of Patterns in Space and Time, Lecture Notes in Biomath., Springer, Berlin, 55 (1984), 61-66.  doi: 10.1007/978-3-642-45589-6_6. [15] A. de Pablo and J. L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equations, 93 (1991), 19-61.  doi: 10.1016/0022-0396(91)90021-Z. [16] A. Ducrot, X. Fu and P. Magal, Turing and Turing-Hopf bifurcations for a reaction diffusion equation with nonlocal advection, J. Nonlinear Sci., 28 (2018), 1959-1997.  doi: 10.1007/s00332-018-9472-z. [17] A. Ducrot and P. Magal, Asymptotic behavior of a nonlocal diffusive logistic equation, [18] A. Ducrot and D. Manceau, A one-dimensional logistic like equation with nonlinear and nonlocal diffusion: Strong convergence to equilibrium, Proc. Amer. Math. Soc., 148 (2020), 3381-3392.  doi: 10.1090/proc/14971. [19] J. Dyson, S. A. Gourley, R. Villella-Bressan and G. F. Webb, Existence and asymptotic properties of solutions of a nonlocal evolution equation modeling cell-cell adhesion, SIAM J. Math. Anal., 42 (2010), 1784-1804.  doi: 10.1137/090765663. [20] R. Eftimie, G. de Vries, M. A. Lewis and F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol., 69 (2007), 1537-1565.  doi: 10.1007/s11538-006-9175-8. [21] X. Fu, Q. Griette and P. Magal, A cell-cell repulsion model on a hyperbolic Keller-Segel equation, J. Math. Biol., 80 (2020), 2257-2300.  doi: 10.1007/s00285-020-01495-w. [22] F. Hamel and C. Henderson, Propagation in a Fisher-KPP equation with non-local advection, J. Funct. Anal., 278 (2020), 108426, 53 pp. doi: 10.1016/j.jfa.2019.108426. [23] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3. [24] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6. [25] S. Katsunuma, H. Honda, T. Shinoda, Y. Ishimoto, T. Miyata, H. Kiyonari, T. Abe, K.-i. Nibu, Y. Takai and H. Togashi, Synergistic action of nectins and cadherins generates the mosaic cellular pattern of the olfactory epithelium, Journal of Cell Biology, 212 (2016), 561-575.  doi: 10.1083/jcb.201509020. [26] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [27] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6. [28] A. J. Leverentz, C. M. Topaz and A. J. Bernoff, Asymptotic dynamics of attractive-repulsive swarms, SIAM J. Appl. Dyn. Syst., 8 (2009), 880-908.  doi: 10.1137/090749037. [29] A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158. [30] A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation, J. Math. Biol., 47 (2003), 353-389.  doi: 10.1007/s00285-003-0209-7. [31] D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66.  doi: 10.1007/s00285-004-0279-1. [32] H. Murakawa and H. Togashi, Continuous models for cell–cell adhesion, Journal of Theoretical Biology, 374 (2015), 1-12.  doi: 10.1016/j.jtbi.2015.03.002. [33] J. Pasquier, P. Magal, C. Boulangé-Lecomte, G. Webb and F. Le Foll, Consequences of cell-to-cell p-glycoprotein transfer on acquired multidrug resistance in breast cancer: a cell population dynamics model, Biol. Direct., 6 (2011), 5. doi: 10.1186/1745-6150-6-5. [34] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407. [35] B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335.  doi: 10.1090/S0002-9947-08-04656-4. [36] W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. [37] Y. Song, S. Wu and H. Wang, Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect, J. Differential Equations, 267 (2019), 6316-6351.  doi: 10.1016/j.jde.2019.06.025. [38] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. [39] E. Zeidler, Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.
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