The Patlak-Keller-Segel model and its variations: Properties of solutions via maximum principle

SIAM Journal on Mathematical Analysis

Volumn 44, Issue 2, 2012, Pages 568-602

  Kim, Inwon ^a   Yao, Yao ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Diffusion aggregation model; Finite speed of propagation; Keller Segel; Long time behavior; Maximum principle

Indexed keywords

ASYMPTOTIC BEHAVIOR OF SOLUTIONS; COMPACTLY-SUPPORTED SOLUTIONS; DIFFUSION-AGGREGATION MODEL; FINITE SPEED OF PROPAGATION; KELLER-SEGEL; LONG TIME BEHAVIOR; MASS CONCENTRATION; NONLOCAL; PATLAK-KELLER-SEGEL MODEL; PROPERTIES OF SOLUTIONS; RADIAL SOLUTIONS; TYPE ARGUMENTS;

ASYMPTOTIC ANALYSIS; DIFFUSION; MAXIMUM PRINCIPLE; NONLINEAR EQUATIONS;

AGGLOMERATION;

EID: 84861372019     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/110823584     Document Type: Article

References (42)

1. [AGS] L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Lectures Math., ETH Zürich, Birkhaüser, Basel, 2008.
2. [B1] J. Bedrossian, Intermediate asymptotics for critical and supercritical aggregation equations and Patlak-Keller-Segel models, arXiv:1009.6187, 2011.
3. [B2] J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, arXiv:1009.5370, 2011.
4. [BRB] J. Bedrossian, N. Rodríguez, and A. L. Bertozzi, Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity, 24 (2011), pp. 1683-1714.
5. [BBC] P. Bénilan, H. Brezis, and M. G. Crandall, A Semilinear Equation in L1(RN), Ann. Scuola Norm. Sup. Pisa, 2 (1975), pp. 523-555.
6. [BH] M. Bertch and D. Hilhorst, A density dependent diffusion equation in population dynamics: Stabilization to equilibrium, SIAM. J. Math. Anal., 17 (1986), pp. 863-882.
7. [BGL] A. L. Bertozzi, J. Garnett, and T. Laurent, Characterization of radially symmetric finite time blowup in multidimensional aggregation equations, SIAM J. Math. Anal., to appear.
8. [BS] A. L. Bertozzi and D. Slepcev, Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion, Commun. Pure. Appl. Anal., 9 (2010), pp. 1617-1637.
9. [BKLN] P. Biler, G. Karch, P. Laurencot, and T. Nadzieja, The 8p-problem for radially symmetric solutions of a chemotaxis model in the plane, M3AS, 29 (2006), pp. 1563-1583. (Pubitemid 44220492)
10. [BCL] A. Blanchet, J. A. Carrillo, and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var., 35 (2009), pp. 133-168.
11. [BlCM] A. Blanchet, J. A. Carrillo, and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in R2, Comm. Pure Appl. Math., 61 (2008), pp. 1449-1481.
12. [BDP] 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), pp. 1-33.
13. [BoCM] S. Boi, V. Capasso, and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens, Nonlinear Anal. Real World Appl., 1 (2000), pp. 163-176.
14. [BuCM] M. Burger, V. Capasso, and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Anal. Real World Appl., 8 (2007), pp. 939-958. (Pubitemid 46097518)
15. [BD] M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media, 3 (2008), pp. 749-785.
16. [BDF] M. Burger, M. Di Francesco, and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, arXiv:1103.5365, 2011.
17. [C] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), pp. 269-361.
18. [CJMTU] J. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani, and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequatlieis, Montash. Math., 133 (2001), pp. 1-82. (Pubitemid 33585637)
19. [CMV] J. A. Carrillo, R. J. McCann, and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), pp. 217-263. (Pubitemid 43237778)
20. [CLW] L. Chen, J.-G. Liu, and J. Wang, Multi-dimensional degenerate Keller-Segel system with critical diffusion exponent 2n/n + 2, SIAM J. Math. Anal., to appear.
21. [CL] M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), pp. 265-298.
22. [Dib] E. Dibenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J., 32 (1983), pp. 83-118.
23. [DP] J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional Keller-Segel model in R2, C. R. Acad. Sci. Paris, 339 (2004), pp. 611-616. (Pubitemid 41447527)
24. [DNR] J. Diaz, T. Nagai, and J. M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotexis system on Rd, J. Differential Equations, 145 (1998), pp. 156-183. (Pubitemid 128347141)
25. [FLP] F. Filbet, P. Laurencot, and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), pp. 189-207. (Pubitemid 40240749)
26. [GM] E. M. Gurtin and R. C McCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), pp. 35-47.
27. [HV] M. A. Herrero and J. L. Velazquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), pp. 177-194. (Pubitemid 126769451)
28. [KL] I. C. Kim and H. K. Lei, Degenerate diffusion with a drift potential: A viscosity solutions approach, Discrete Contin. Dyn. Syst., 27 (2010), pp. 767-786.
29. [KS] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), pp. 225-234.
30. [LSU] O. A. Ladyzhenskaia, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. (23), American Mathematical Society, Providence, RI, 1967.
31. [LL] I. R. Lapidus and M. Levandowsky, Modeling chemosensory responses of swimming eukaryotes, in Biological Growth and Spread (Proc. Conf., Heidelberg, 1979), Lecture Notes in Biomath. 28, Springer, Berlin, 1980, pp. 388-396.
32. [LS1] S. Luckhaus and Y. Sugiyama, Large time behavior of solutions in super-critical case to degenerate Keller-Segel systems, Math. Model. Numer. Anal., 40 (2006), pp. 597-621. (Pubitemid 44106970)
33. [LS2] S. Luckhaus and Y. Sugiyama, Asymptotic profile with optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases, Indiana Univ. Math. J., 56 (2007), pp. 1279-1297.
34. [LY] E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), pp. 147-174.
35. [L] P. L. Lions, The concentration-compactness principle in calculus of variations. The locally compact case, Part 1, Ann. Inst. H. Poincaré, 1 (1984), pp. 109-145.
36. [P] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), pp. 311-338.
37. [PV] Perthame and Vassuer, Regularization in Keller-Segel type systems and the De Giorgi method, Comm. Math. Sc., 10 (2012), pp. 463-476.
38. [S1] Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in supercritical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), pp. 841-876.
39. [S2] Y. Sugiyama, The global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), pp. 133-180.
40. [TBL] C. M. Topaz, A. L. Bertozzi, and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), pp. 1601-1623. (Pubitemid 44392962)
41. [V] J. Vazquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, Oxford, UK, 2007.
42. [Vi] C. Villani, Optimal transportation, dissipative PDEs and functional in-equalities, in Optimal Transportation and Applications (Martina Franca, 2001), Lecture Notes in Math. 1813, Springer, Berlin, 2003, pp. 53-89.