On a repulsion-diffusion equation with immigration

Peter Koepernik

doi:10.3934/dcds.2023140

Article Contents

2024, Volume 44, Issue 4: 1106-1133. Doi: 10.3934/dcds.2023140

This issue Previous Article A class of critical $ p $-Kirchhoff type equations on closed manifolds Next Article Stress blow-up analysis when suspending rigid particles approach boundary in 3D Stokes flow

On a repulsion-diffusion equation with immigration

Peter Koepernik

Department of Statistics, University of Oxford, 24–29 St Giles', Oxford, OX1 3LB, United Kingdom

Received: April 2023

Revised: September 2023

Early access: November 21, 2023

Published online: November 21, 2023

The author is supported by EPSRC grant EP/W523781/1.

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Abstract

We study a repulsion-diffusion equation with immigration and linear diffusion, whose asymptotic behaviour is related to stability of long-term dynamics in spatial population models and other branching particle systems. We prove well-posedness and find sharp conditions on the repulsion under which a form of the maximum principle and a strong notion of global boundedness of solutions hold. The critical asymptotic strength of the repulsion is $ |x|^{1-d} $, that of the Newtonian potential.

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Mathematics Subject Classification: Primary: 35B40, 35B50; Secondary: 35Q92, 60J68.

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Figure 1. Evolution of the upper bound on $ \left\|\rho_t\right\|_\infty $ in Corollary 3.9 for $ c_W > 0 $ and different values of $ \left\|\rho_0\right\|_\infty $

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[1]	E. Aïdékon, J. Berestycki, É. Brunet and Z. Shi, Branching Brownian motion seen from its tip, Probab. Theory and Related Fields, 157 (2013), 405-451. doi: 10.1007/s00440-012-0461-0.
[2]	L.-P. Arguin, A. Bovier and N. Kistler, The extremal process of branching Brownian motion, Probab. Theory Related Fields, 157 (2013), 535-574. doi: 10.1007/s00440-012-0464-x.
[3]	J. Bedrossian and N. Rodriguez, Inhomogeneous Patlak–Keller–Segel models and aggregation equations with nonlinear diffusion in ${R}^d$, Discrete Contin. Dyn. Syst., Ser. B, 19 (2014), 1279-1309. doi: 10.3934/dcdsb.2014.19.1279.
[4]	J. Bedrossian, N. Rodriguez and A. L. Bertozzi, Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion, Nonlinearity, 24 (2011), 1683-1714. doi: 10.1088/0951-7715/24/6/001.
[5]	A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, (2006), No. 44, 32 pp.
[6]	A. Bovier and L. Hartung, Branching Brownian motion with self-repulsion, in Ann. Henri Poincaré, 24 (2023), 931-956. doi: 10.1007/s00023-022-01223-8.
[7]	L. Brasco, D. Gómez-Castro and J. L. Vázquez, Characterisation of homogeneous fractional Sobolev spaces, Calc. Var. Partial Differ. Equ., 60 (2021), Paper No. 60, 40 pp. doi: 10.1007/s00526-021-01934-6.
[8]	M. Burger, M. Di Francesco and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Commun. Math. Sci., 11 (2013), 709-738. doi: 10.4310/CMS.2013.v11.n3.a3.
[9]	V. Calvez and J. A. Carrillo, Volume effects in the Keller–Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. (9), 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002.
[10]	J. F. Carlson and J. R. Oppenheimer, On multiplicative showers, Phys. Rev., 51 (1937), 220. doi: 10.1103/PhysRev.51.220.
[11]	J. A. Carrillo, K. Craig and Y. Yao, Aggregation-diffusion equations: Dynamics, asymptotics, and singular limits, Active Particles, Volume 2, 65-108. doi: 10.1007/978-3-030-20297-2_3.
[12]	J. A. Carrillo, D. Gómez-Castro, Y. Yao and C. Zeng, Asymptotic simplification of aggregation-diffusion equations towards the heat kernel, Arch. Ration. Mech. Anal., 247 (2023), Paper No. 11, 45 pp. doi: 10.1007/s00205-022-01838-5.
[13]	J. A. Carrillo, S. Hittmeir, B. Volzone and Y. Yao, Nonlinear aggregation-diffusion equations: Radial symmetry and long time asymptotics, Invent. Math., 218 (2019), 889-977. doi: 10.1007/s00222-019-00898-x.
[14]	J. A. Carrillo, F. Hoffmann, E. Mainini and B. Volzone, Ground states in the diffusion-dominated regime, Calc. Var. Partial Differ. Equ., 57 (2018), 1-28. doi: 10.1007/s00526-018-1402-2.
[15]	L. Chen, J.-G. Liu and J. Wang, Multidimensional degenerate Keller–Segel system with critical diffusion exponent 2n/(n+2), SIAM J. Math. Anal., 44 (2012), 1077-1102. doi: 10.1137/110839102.
[16]	L. Chen and J. Wang, Exact criterion for global existence and blow up to a degenerate Keller–Segel system, Doc. Math., 19 (2014), 103-120. doi: 10.4171/dm/441.
[17]	D. A. Dawson, Measure-Valued Markov Processes, Springer, 1993. doi: 10.1007/BFb0084190.
[18]	F. del Teso, D. Gómez-Castro and J. L. Vázquez, Estimates on translations and Taylor expansions in fractional Sobolev spaces, Nonlinear Anal., 200 (2020), 111995. doi: 10.1016/j.na.2020.111995.
[19]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[20]	M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302. doi: 10.1103/PhysRevLett.96.104302.
[21]	E. B. Dynkin, Branching particle systems and superprocesses, Ann. Probab., 19 (1991), 1157-1194.
[22]	J. Engländer, The center of mass for spatial branching processes and an application for self-interaction, Electronic J. Probab., 15 (2010), 1938-1970. doi: 10.1214/EJP.v15-822.
[23]	J. Engländer and L. Zhang, Branching diffusion with particle interactions, Electron. J. Probab., 21 (2016), 1-25. doi: 10.1214/16-EJP4782.
[24]	A. M. Etheridge, An Introduction to Superprocesses, 20, American Mathematical Soc., 2000. doi: 10.1090/ulect/020.
[25]	A. M. Etheridge, Survival and extinction in a locally regulated population, Ann. Appl. Probab., 14 (2004), 188-214. doi: 10.1214/aoap/1075828051.
[26]	S. N. Evans, Two representations of a conditioned superprocess, Proc. R. Soc. Edinb., Sect. A, Math., 123 (1993), 959-971. doi: 10.1017/S0308210500029619.
[27]	J. Felsenstein, A pain in the torus: Some difficulties with models of isolation by distance, The American Naturalist, 109 (1975), 359-368, URL http://www.jstor.org/stable/2459700. doi: 10.1086/283003.
[28]	H. Gill, A Super Onstein-Uhlenbeck process interacting with its center of mass, Ann. Probab., 41 (2013), 989-1029. doi: 10.1214/11-AOP741.
[29]	M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623. doi: 10.1007/BF01445268.
[30]	W.-M. Hong and Z.-H. Li, A central limit theorem for super-Brownian motion with super-Brownian immigration, J. Appl. Probab., 36 (1999), 1218-1224. doi: 10.1239/jap/1032374767.
[31]	D. Horstmann, From 1970 until present: The Keller–Segel model in chemotaxis and its consequences.,
[32]	P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinet. Relat. Models, 7 (2014), 661-711. doi: 10.3934/krm.2014.7.661.
[33]	P.-E. Jabin and Z. Wang, Mean field limit and propagation of chaos for Vlasov systems with bounded forces, J. Funct. Anal., 271 (2016), 3588-3627. doi: 10.1016/j.jfa.2016.09.014.
[34]	G. Kaib, Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential, SIAM J. Math. Anal., 49 (2017), 272-296. doi: 10.1137/16M1072450.
[35]	O. Kallenberg, Stability of critical cluster fields, Math. Nachr., 77 (1977), 7-43. doi: 10.1002/mana.19770770102.
[36]	E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.
[37]	Z.-H. Li, Measure-valued branching processes with immigration, Stochastic Process. Appl., 43 (1992), 249-264. doi: 10.1016/0304-4149(92)90061-T.
[38]	Z.-H. Li, Immigration structures associated with Dawson-Watanabe superprocesses, Stochastic Process. Appl., 62 (1996), 73-86. doi: 10.1016/0304-4149(95)00087-9.
[39]	E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174.
[40]	K. Oelschläger, Large systems of interacting particles and the porous medium equation, J. Differ. Equations, 88 (1990), 294-346. doi: 10.1016/0022-0396(90)90101-T.
[41]	C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biol., 15 (1953), 311-338. doi: 10.1007/bf02476407.
[42]	E. Perkins, Dawson-Watanabe superprocesses and measure-valued diffusions, in Lectures on Probability Theory and Statistics (Saint-Flour, 1999), vol. 1781 of Lecture Notes in Math., Springer, Berlin, 2002,125-324.
[43]	Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differ. Integral Equ., 20 (2007), 133-180.
[44]	C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6.
[45]	H. R. Vega-Carrillo, I. R. Esparza-Garcia and A. Sanchez, Features of a subcritical nuclear reactor, Annals of Nuclear Energy, 75 (2015), 101-106. doi: 10.1016/j.anucene.2014.08.006.
[46]	H. Wenming, Limiting behavior of the super-Brownian motion with super-Brownian immigration, CR Math. Rep. Acad. Sci. Canada, 25 (2003), 1-6.
[47]	M. Zhang, Large deviations for super-Brownian motion with immigration, J. Appl. Probab., 41 (2004), 187-201. doi: 10.1239/jap/1077134677.
[48]	M. Zhang, Some scaled limit theorems for an immigration super-Brownian motion, Science in China Series A: Mathematics, 51 (2008), 203-214. doi: 10.1007/s11425-007-0176-3.
[49]	Y. P. Zhang, On a class of diffusion-aggregation equations, Discrete Contin. Dyn. Syst., 40 (2020), 907-932. doi: 10.3934/dcds.2020066.