# American Institute of Mathematical Sciences

July  2014, 19(5): 1279-1309. doi: 10.3934/dcdsb.2014.19.1279

## Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $\mathbb{R}^d$

 1 New York University, Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012-1185, United States 2 Stanford University, Department of Mathematics, Building 380, Sloan Hall Stanford, CA 94305, United States

Received  September 2011 Revised  March 2012 Published  April 2014

Aggregation equations and parabolic-elliptic Patlak-Keller-Segel (PKS) systems for chemotaxis with nonlinear diffusion are popular models for nonlocal aggregation phenomenon and are a source of many interesting mathematical problems in nonlinear PDEs. The purpose of this work is to give a more complete study of local, subcritical and small-data critical/supercritical theory in $\mathbb{R}^d$, $d \geq 2$. Some existing results can be found in the literature; however, one of the most important cases in biological applications, that is the $\mathbb{R}^2$ case, had not been studied. In this paper, we treat two related systems, which are different generalizations of the classical parabolic-elliptic PKS model. In the first class, nonlocal aggregation is modeled by convolution with a general interaction potential, studied in this generality in our previous work [6]. For this class of models we also present several large data global existence results for critical problems. The second class is a variety of PKS models with spatially inhomogeneous diffusion and decay rate of the chemo-attractant, which is potentially relevant to biological applications and raises interesting mathematical questions.
Citation: Jacob Bedrossian, Nancy Rodríguez. Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $\mathbb{R}^d$. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1279-1309. doi: 10.3934/dcdsb.2014.19.1279
##### References:
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Anal., 45 (2013), 934-964. arXiv:1108.5301. doi: 10.1137/120882731.  Google Scholar [6] J. Bedrossian, N. Rodríguez and A. 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.  Google Scholar [7] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 48 of Colloquium Publications, American Mathematical Society, 2000.  Google Scholar [8] A. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Comm. Math. Sci., 8 (2010), 45-65. doi: 10.4310/CMS.2010.v8.n1.a4.  Google Scholar [9] A. Bertozzi and D. Slepčev, Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion, Comm. Pure. Appl. Anal., 9 (2010), 1617-1637. doi: 10.3934/cpaa.2010.9.1617.  Google Scholar [10] P. Biler and T. Nadzieja, Global and exploding solutions in a model of self-gravitating systems, Reports on Mathematical Physics, 52 (2003), 205-225, URL http://www.sciencedirect.com/science/article/pii/S0034487703900139. doi: 10.1016/S0034-4877(03)90013-9.  Google Scholar [11] A. Blanchet, On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher,, , ().   Google Scholar [12] A. Blanchet, V. Calvez and J. Carrillo, Convergence of the mass-transport steepest descent scheme for subcritical Patlak-Keller-Segel model, SIAM J. Num. Anal., 46 (2008), 691-721. doi: 10.1137/070683337.  Google Scholar [13] A. Blanchet, E. Carlen and J. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, Journal of Functional Analysis, 262 (2012), 2142-2230. arXiv:1009.0134. doi: 10.1016/j.jfa.2011.12.012.  Google Scholar [14] A. Blanchet, J. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var., 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7.  Google Scholar [15] A. Blanchet, J. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbb R^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481. doi: 10.1002/cpa.20225.  Google Scholar [16] A. Blanchet, J. Dolbeault, M. Escobedo and J. Fernández, Asymptotic behavior for small mass in the two-dimensional parabolic-elliptic Keller-Segel model, J. Math. Anal. Appl., 361 (2010), 533-542. doi: 10.1016/j.jmaa.2009.07.034.  Google Scholar [17] A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, E. J. Diff. Eqn, 2006 (2006), 1-32.  Google Scholar [18] 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), 163-176, Spatial heterogeneity in ecological models (Alcalá de Henares, 1998). doi: 10.1016/S0362-546X(99)00399-5.  Google Scholar [19] M. Brenner, P. Constantin, L. Kadanoff, A. Schenkel and S. Venkataramani, Diffusion, attraction and collapse, Nonlinearity, 12 (1999), 1071-1098. doi: 10.1088/0951-7715/12/4/320.  Google Scholar [20] M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlin. Anal. Real World Appl., 8 (2007), 939-958. doi: 10.1016/j.nonrwa.2006.04.002.  Google Scholar [21] M. Burger, M. D. Francesco and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Communications in Mathematical Sciences, 11 (2013), 709-738. arXiv:1103.5365. doi: 10.4310/CMS.2013.v11.n3.a3.  Google Scholar [22] V. Calvez and J. Carrillo, Volume effects in the {Keller-Segel} model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002.  Google Scholar [23] E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $\mathbb S^n$, Geom. Func. Anal., 2 (1992), 90-104. doi: 10.1007/BF01895706.  Google Scholar [24] J. Carrillo, A. Jüngel, P. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Montash. Math., 133 (2001), 1-82. doi: 10.1007/s006050170032.  Google Scholar [25] P. Chavanis, J. Sommeria and R. Robert, Statistical mechanics of two-dimensional vortices and collisionless stellar systems, The Astrophysical Journal, 471 (1996), 385. doi: 10.1086/177977.  Google Scholar [26] L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x.  Google Scholar [27] J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional Keller-Segel model in $\mathbb R^2$, C.R. Acad. Sci. Paris, Sér I Math, 339 (2004), 611-616. doi: 10.1016/j.crma.2004.08.011.  Google Scholar [28] L. Evans, Partial Differential Equations, vol. 19 of Grad. Stud. Math., American Mathematical Society, 1998. Google Scholar [29] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, 2001.  Google Scholar [30] E. M. Gurtin and R. McCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1.  Google Scholar [31] 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.  Google Scholar [32] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein, 105 (2003), 103-165.  Google Scholar [33] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differntial equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.2307/2153966.  Google Scholar [34] E. F. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. Google Scholar [35] R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity, Amer. Math. Soc., Providence, RI, Clay Math. Proc.,, Evolution equations, 17 (2013), 325-437.  Google Scholar [36] I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2), (2012), 568-602 Google Scholar [37] R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588. doi: 10.1016/j.jmaa.2004.12.009.  Google Scholar [38] E. Lieb and M. Loss, Analysis, vol. 14 of Grad. Stud. Math., American Mathematical Society, 2001.  Google Scholar [39] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996.  Google Scholar [40] P. Lions, The concentration-compactness principle in the calculus of variations. the locally compact case, part 1, Ann. I.H.P., Anal. Nonlin., 1 (1984), 109-145.  Google Scholar [41] 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), 597-621. doi: 10.1051/m2an:2006025.  Google Scholar [42] 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), 1279-1297. doi: 10.1512/iumj.2007.56.2977.  Google Scholar [43] P. A. Milewski and X. Yang, A simple model for biological aggregation with asymmetric sensing, Comm. Math. Sci., 6 (2008), 397-416. doi: 10.4310/CMS.2008.v6.n2.a7.  Google Scholar [44] A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, Journal of Mathematical Biology, 38 (1999), 534-570. doi: 10.1007/s002850050158.  Google Scholar [45] D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, Journal of mathematical biology, 50 (2005), 49-66. doi: 10.1007/s00285-004-0279-1.  Google Scholar [46] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.  Google Scholar [47] A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds, Advances in Biophysics, 22 (1986), 1-94. doi: 10.1016/0065-227X(86)90003-1.  Google Scholar [48] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Part. Diff. Eqn., 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar [49] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.  Google Scholar [50] B. Perthame and A. Vasseur, Regularization in Keller-Segel type systems and the De Giorgi method, Commun. Math. Sci., 10 (2012), 463-476. doi: 10.4310/CMS.2012.v10.n2.a2.  Google Scholar [51] R. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, vol. 49 of Math. Surveys and Monographs, American Mathematical Society, 1997.  Google Scholar [52] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993.  Google Scholar [53] Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Diff. Int. Eqns., 19 (2006), 841-876.  Google Scholar [54] Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Diff. Eqns., 12 (2007), 121-144.  Google Scholar [55] Y. Sugiyama, The global existence and asymptotic behavior of solutions to degenerate to quasi-linear parabolic systems of chemotaxis, Diff. Int. Eqns., 20 (2007), 133-180.  Google Scholar [56] C. Topaz and A. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174. doi: 10.1137/S0036139903437424.  Google Scholar [57] C. Topaz, A. Bertozzi and M. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Bio, 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6.  Google Scholar [58] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.   Google Scholar [59] T. Witelski, A. Bernoff and A. Bertozzi, Blowup and dissipation in a critical-case unstable thin film equation, Euro. J. Appl. Math., 15 (2004), 223-256. doi: 10.1017/S0956792504005418.  Google Scholar

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##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaŕe, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, Birkhäuser, 2005.  Google Scholar [2] J. Azzam and J. Bedrossian, Bounded mean oscillation and the uniqueness of active scalars,, To appear in Trans. Amer. Math. Soc., ().   Google Scholar [3] J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Appl. Math. Letters, 24 (2011), 1927-1932. doi: 10.1016/j.aml.2011.05.022.  Google Scholar [4] J. Bedrossian, Intermediate asymptotics for critical and supercritical aggregation equations and Patlak-Keller-Segel models, Comm. Math. Sci., 9 (2011), 1143-1161. doi: 10.4310/CMS.2011.v9.n4.a11.  Google Scholar [5] J. Bedrossian and I. Kim, Global existence and finite time blow-up for critical Patlak-Keller-Segel models with inhomogeneous diffusion, SIAM J. Math. Anal., 45 (2013), 934-964. arXiv:1108.5301. doi: 10.1137/120882731.  Google Scholar [6] J. Bedrossian, N. Rodríguez and A. 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.  Google Scholar [7] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 48 of Colloquium Publications, American Mathematical Society, 2000.  Google Scholar [8] A. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Comm. Math. Sci., 8 (2010), 45-65. doi: 10.4310/CMS.2010.v8.n1.a4.  Google Scholar [9] A. Bertozzi and D. Slepčev, Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion, Comm. Pure. Appl. Anal., 9 (2010), 1617-1637. doi: 10.3934/cpaa.2010.9.1617.  Google Scholar [10] P. Biler and T. Nadzieja, Global and exploding solutions in a model of self-gravitating systems, Reports on Mathematical Physics, 52 (2003), 205-225, URL http://www.sciencedirect.com/science/article/pii/S0034487703900139. doi: 10.1016/S0034-4877(03)90013-9.  Google Scholar [11] A. Blanchet, On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher,, , ().   Google Scholar [12] A. Blanchet, V. Calvez and J. Carrillo, Convergence of the mass-transport steepest descent scheme for subcritical Patlak-Keller-Segel model, SIAM J. Num. Anal., 46 (2008), 691-721. doi: 10.1137/070683337.  Google Scholar [13] A. Blanchet, E. Carlen and J. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, Journal of Functional Analysis, 262 (2012), 2142-2230. arXiv:1009.0134. doi: 10.1016/j.jfa.2011.12.012.  Google Scholar [14] A. Blanchet, J. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var., 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7.  Google Scholar [15] A. Blanchet, J. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbb R^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481. doi: 10.1002/cpa.20225.  Google Scholar [16] A. Blanchet, J. Dolbeault, M. Escobedo and J. Fernández, Asymptotic behavior for small mass in the two-dimensional parabolic-elliptic Keller-Segel model, J. Math. Anal. Appl., 361 (2010), 533-542. doi: 10.1016/j.jmaa.2009.07.034.  Google Scholar [17] A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, E. J. Diff. Eqn, 2006 (2006), 1-32.  Google Scholar [18] 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), 163-176, Spatial heterogeneity in ecological models (Alcalá de Henares, 1998). doi: 10.1016/S0362-546X(99)00399-5.  Google Scholar [19] M. Brenner, P. Constantin, L. Kadanoff, A. Schenkel and S. Venkataramani, Diffusion, attraction and collapse, Nonlinearity, 12 (1999), 1071-1098. doi: 10.1088/0951-7715/12/4/320.  Google Scholar [20] M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlin. Anal. Real World Appl., 8 (2007), 939-958. doi: 10.1016/j.nonrwa.2006.04.002.  Google Scholar [21] M. Burger, M. D. Francesco and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Communications in Mathematical Sciences, 11 (2013), 709-738. arXiv:1103.5365. doi: 10.4310/CMS.2013.v11.n3.a3.  Google Scholar [22] V. Calvez and J. Carrillo, Volume effects in the {Keller-Segel} model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002.  Google Scholar [23] E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $\mathbb S^n$, Geom. Func. Anal., 2 (1992), 90-104. doi: 10.1007/BF01895706.  Google Scholar [24] J. Carrillo, A. Jüngel, P. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Montash. Math., 133 (2001), 1-82. doi: 10.1007/s006050170032.  Google Scholar [25] P. Chavanis, J. Sommeria and R. Robert, Statistical mechanics of two-dimensional vortices and collisionless stellar systems, The Astrophysical Journal, 471 (1996), 385. doi: 10.1086/177977.  Google Scholar [26] L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x.  Google Scholar [27] J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional Keller-Segel model in $\mathbb R^2$, C.R. Acad. Sci. Paris, Sér I Math, 339 (2004), 611-616. doi: 10.1016/j.crma.2004.08.011.  Google Scholar [28] L. Evans, Partial Differential Equations, vol. 19 of Grad. Stud. Math., American Mathematical Society, 1998. Google Scholar [29] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, 2001.  Google Scholar [30] E. M. Gurtin and R. McCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1.  Google Scholar [31] 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.  Google Scholar [32] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein, 105 (2003), 103-165.  Google Scholar [33] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differntial equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.2307/2153966.  Google Scholar [34] E. F. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. Google Scholar [35] R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity, Amer. Math. Soc., Providence, RI, Clay Math. Proc.,, Evolution equations, 17 (2013), 325-437.  Google Scholar [36] I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2), (2012), 568-602 Google Scholar [37] R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588. doi: 10.1016/j.jmaa.2004.12.009.  Google Scholar [38] E. Lieb and M. Loss, Analysis, vol. 14 of Grad. Stud. Math., American Mathematical Society, 2001.  Google Scholar [39] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996.  Google Scholar [40] P. Lions, The concentration-compactness principle in the calculus of variations. the locally compact case, part 1, Ann. I.H.P., Anal. Nonlin., 1 (1984), 109-145.  Google Scholar [41] 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), 597-621. doi: 10.1051/m2an:2006025.  Google Scholar [42] 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), 1279-1297. doi: 10.1512/iumj.2007.56.2977.  Google Scholar [43] P. A. Milewski and X. Yang, A simple model for biological aggregation with asymmetric sensing, Comm. Math. Sci., 6 (2008), 397-416. doi: 10.4310/CMS.2008.v6.n2.a7.  Google Scholar [44] A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, Journal of Mathematical Biology, 38 (1999), 534-570. doi: 10.1007/s002850050158.  Google Scholar [45] D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, Journal of mathematical biology, 50 (2005), 49-66. doi: 10.1007/s00285-004-0279-1.  Google Scholar [46] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.  Google Scholar [47] A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds, Advances in Biophysics, 22 (1986), 1-94. doi: 10.1016/0065-227X(86)90003-1.  Google Scholar [48] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Part. Diff. Eqn., 26 (2001), 101-174. doi: 10.1081/PDE-100002243.  Google Scholar [49] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. 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