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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J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion,, Appl. Math. Letters, 24 (2011), 1927. doi: 10.1016/j.aml.2011.05.022. Google Scholar

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J. Bedrossian, Intermediate asymptotics for critical and supercritical aggregation equations and Patlak-Keller-Segel models,, Comm. Math. Sci., 9 (2011), 1143. doi: 10.4310/CMS.2011.v9.n4.a11. Google Scholar

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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. doi: 10.1137/120882731. Google Scholar

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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. doi: 10.1088/0951-7715/24/6/001. Google Scholar

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Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 48 of Colloquium Publications,, American Mathematical Society, (2000). Google Scholar

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A. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation,, Comm. Math. Sci., 8 (2010), 45. 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. doi: 10.3934/cpaa.2010.9.1617. Google Scholar

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P. Biler and T. Nadzieja, Global and exploding solutions in a model of self-gravitating systems,, Reports on Mathematical Physics, 52 (2003), 205. doi: 10.1016/S0034-4877(03)90013-9. Google Scholar

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A. Blanchet, On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher,, , (). Google Scholar

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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. doi: 10.1137/070683337. Google Scholar

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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. doi: 10.1016/j.jmaa.2009.07.034. Google Scholar

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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. Google Scholar

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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. doi: 10.1016/S0362-546X(99)00399-5. Google Scholar

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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. 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. 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. 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. 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. doi: 10.1007/s006050170032. Google Scholar

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P. Chavanis, J. Sommeria and R. Robert, Statistical mechanics of two-dimensional vortices and collisionless stellar systems,, The Astrophysical Journal, 471 (1996). doi: 10.1086/177977. Google Scholar

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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. 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, 339 (2004), 611. 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, (2001). Google Scholar

[30]

E. M. Gurtin and R. McCamy, On the diffusion of biological populations,, Math. Biosci., 33 (1977), 35. 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. 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, 105 (2003), 103. 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. doi: 10.2307/2153966. Google Scholar

[34]

E. F. Keller and L. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. Google Scholar

[35]

R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity,, Amer. Math. Soc., 17 (2013), 325. 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 (2012), 568. Google Scholar

[37]

R. Kowalczyk, Preventing blow-up in a chemotaxis model,, J. Math. Anal. Appl., 305 (2005), 566. 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., (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., 1 (1984), 109. 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. 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. 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. 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. 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. 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. Google Scholar

[47]

A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds,, Advances in Biophysics, 22 (1986), 1. 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. doi: 10.1081/PDE-100002243. Google Scholar

[49]

C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. 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. 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. 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. 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. 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. 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. 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. doi: 10.1017/S0956792504005418. Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaŕe, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, Lectures in Mathematics, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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, 339 (2004), 611. 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, (2001). Google Scholar

[30]

E. M. Gurtin and R. McCamy, On the diffusion of biological populations,, Math. Biosci., 33 (1977), 35. 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. 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, 105 (2003), 103. 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. doi: 10.2307/2153966. Google Scholar

[34]

E. F. Keller and L. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. Google Scholar

[35]

R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity,, Amer. Math. Soc., 17 (2013), 325. 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 (2012), 568. Google Scholar

[37]

R. Kowalczyk, Preventing blow-up in a chemotaxis model,, J. Math. Anal. Appl., 305 (2005), 566. 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., (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., 1 (1984), 109. 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. 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. 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. 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. 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. 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. Google Scholar

[47]

A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds,, Advances in Biophysics, 22 (1986), 1. 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. doi: 10.1081/PDE-100002243. Google Scholar

[49]

C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. 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. 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. 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. 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. 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. 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. 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. doi: 10.1017/S0956792504005418. Google Scholar

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