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February  2016, 36(2): 1061-1084. doi: 10.3934/dcds.2016.36.1061

Parabolic elliptic type Keller-Segel system on the whole space case

1. 

School of Mathematics, Liaoning University, Shenyang 110036, China, China

2. 

Universität Mannheim, Lehrstuhl für Mathematik IV, 68131, Mannheim

Received  July 2014 Published  August 2015

This note is devoted to the discussion on the existence and blow up of the solutions to the parabolic elliptic type Patlak-Keller-Segel system on the whole space case. The problem in two dimension is closely related to the Logarithmic Hardy-Littlewood-Sobolev inequality, which directly introduced the critical mass $8\pi$. While in the higher dimension case, it is related to the Hardy-Littlewood-Sobolev inequality. Therefore, a porous media type nonlinear diffusion has been introduced in order to balance the aggregation. We will review the critical exponents which were introduced in the literature, namely, the exponent $m=2-2/n$ which comes from the scaling invariance of the mass, and the exponent $m=2n/(n+2)$ which comes from the conformal invariance of the entropy. Finally a new result on the model with a general potential, inspired from the Hardy-Littlewood-Sobolev inequality, will be given.
Citation: Jinhuan Wang, Li Chen, Liang Hong. Parabolic elliptic type Keller-Segel system on the whole space case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1061-1084. doi: 10.3934/dcds.2016.36.1061
References:
[1]

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.

[2]

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.

[3]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality,, Ann. Math., 138 (1993), 213. doi: 10.2307/2946638.

[4]

S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m\geq 0$,, Comm. Math. Phys., 323 (2013), 1017. doi: 10.1007/s00220-013-1777-z.

[5]

P. Biler and T. Nadzieja, A class of nonlocal parabolic problems occurring in statistical mechanics,, Colloq. Math., 66 (1993), 131.

[6]

A. Blanchet, E. A. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model,, J. Funct. Anal., 262 (2012), 2142. doi: 10.1016/j.jfa.2011.12.012.

[7]

A. Blanchet, On the Parabolic-Elliptic Patlak-Keller-Segel System in Dimension 2 and Higher,, Séminaire Laurent Schwartz$-$EDP et applications, (): 1.

[8]

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), 133. doi: 10.1007/s00526-008-0200-7.

[9]

A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbbR^2$,, Comm. Pure Appl. Math., 61 (2008), 1449. doi: 10.1002/cpa.20225.

[10]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,, Electron. J. Diff. Eqns., 44 (2006).

[11]

A. Blanchet and P. Laurençot, Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion,, Comm. Pure Appl. Math., 11 (2012), 47.

[12]

L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271. doi: 10.1002/cpa.3160420304.

[13]

E. Carlen, J. A. Carrillo and M. Loss, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows,, Proc. Nat. Acad. USA, 107 (2010), 19696. doi: 10.1073/pnas.1008323107.

[14]

E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $S^n$,, Geom. Funct. Anal., 2 (1992), 90. doi: 10.1007/BF01895706.

[15]

J. A. Carrillo, L. Chen, J.-G. Liu and J. Wang, A note on the subcritical two dimensional Keller-Segel system,, Acta appl. math., 119 (2012), 43. doi: 10.1007/s10440-011-9660-4.

[16]

L. Chen, J.-G. Liu and J. H. Wang, Multidimensional degenerate Keller-Segel system with critical diffusion exponent $2n/(n+2)$,, SIAM J. Math. Anal., 44 (2012), 1077. doi: 10.1137/110839102.

[17]

L. Chen and J. H. Wang, Exact criterion for global existence and blow-up to a degenerate Keller-Segel system,, Doc. Math., 19 (2014), 103.

[18]

X. Chen, A. Jüngel and J.-G. Liu, A note on Aubin-Lions-Dubinskii lemmas,, Acta Appl. Math., 133 (2014), 33. doi: 10.1007/s10440-013-9858-8.

[19]

W. X. Chen and C. M. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. Journal, 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8.

[20]

S. Childress, Chemotactic collapse in two dimensions,, Lect. Notes in Biomathematics, 55 (1984), 61. doi: 10.1007/978-3-642-45589-6_6.

[21]

T. Cieślak and P. Laurencot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchwski-Poisson system,, C. R. Acad. Sci. Paris Ser. I, 347 (2009), 237. doi: 10.1016/j.crma.2009.01.016.

[22]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, Nonlinearity, 21 (2008), 1057. doi: 10.1088/0951-7715/21/5/009.

[23]

J. M. Delort, Existence de nappes de tourbillon en dimension deux,, J. Amer. Math. Soc., 4 (1991), 553. doi: 10.1090/S0894-0347-1991-1102579-6.

[24]

R. DiPerna and A. Majda, Concentrations in regularizations for 2-D incompressible flow,, Comm. Pure Appl. Math., 40 (1987), 301. doi: 10.1002/cpa.3160400304.

[25]

J. Dolbeault, Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion,, Math. Res. Lett., 18 (2011), 1037. doi: 10.4310/MRL.2011.v18.n6.a1.

[26]

J. Dolbeault, M. J. Esteban and G. Jankowiak, The Moser-Trudinger-Onofri inequality, preprint,, , ().

[27]

J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional Keller-Segel model in $\mathbbR^2$,, C. R. Acad. Sci. Paris, 339 (2004), 611. doi: 10.1016/j.crma.2004.08.011.

[28]

R. L. Frank and E. H. Lieb, A new rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, Spectral theory, function spaces and inequalities,, Oper. Theory Adv. Appl., 219 (2012), 55. doi: 10.1007/978-3-0348-0263-5_4.

[29]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u + u^{1+\alpha}$,, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109.

[30]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77. doi: 10.1002/mana.19981950106.

[31]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure and Appl. Math., 34 (1981), 525. doi: 10.1002/cpa.3160340406.

[32]

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.

[33]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences I,, Jahresberichte der DMV, 105 (2003), 103.

[34]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences II,, Jahresberichte der DMV, 106 (2004), 51.

[35]

D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model,, J. Math. Biol., 44 (2002), 463. doi: 10.1007/s002850100134.

[36]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022.

[37]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, J. Differential Equations, 252 (2012), 1421. doi: 10.1016/j.jde.2011.02.012.

[38]

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. doi: 10.1090/S0002-9947-1992-1046835-6.

[39]

J. Jang, Nonlinear instability in gravitational Euler-Poisson systems for $\gamma=6/5$,, Arch. Rational Mech. Anal., 188 (2008), 265. doi: 10.1007/s00205-007-0086-0.

[40]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5.

[41]

E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theoret. Biol., 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6.

[42]

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. doi: 10.1137/110823584.

[43]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model,, J. Math. Anal. Appl., 343 (2008), 379. doi: 10.1016/j.jmaa.2008.01.005.

[44]

E. H. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities,, Ann. Math., 118 (1983), 349. doi: 10.2307/2007032.

[45]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14,, $2^{nd}$ edition, (2001). doi: 10.1090/gsm/014.

[46]

J.-G. Liu and Z. P. Xin, Convergence of Vortex Methods for Weak Solution to the 2-D Euler Equations with Vortex Sheet Data,, Comm. Pure Appl. Math., 48 (1995), 611. doi: 10.1002/cpa.3160480603.

[47]

J.-G. Liu and Z. P. Xin, Convergence of point vortex method for 2-D vortex sheet,, Math. Comp., 70 (2001), 595. doi: 10.1090/S0025-5718-00-01271-0.

[48]

S. Luckhaus and Y. Sugiyama, Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems,, M2AN Math. Model. Numer. Anal., 40 (2006), 597. doi: 10.1051/m2an:2006025.

[49]

A. J. Majda, Remarks on Weak Solution for Vortex Sheets with a Distinguished Sign,, Indiana Univ. Math. J., 42 (1993), 921. doi: 10.1512/iumj.1993.42.42043.

[50]

T. Nagai and T. Senba, Global existence and blow-up of radial solutioins to a parabolic-elliptic system of chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 145.

[51]

K. J. Painter and T. Hillen, Volume filling and quorum sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501.

[52]

C. S. Patlak, Random walk with persistenc and external bias,, Bull. Math. Biol. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407.

[53]

B. Perthame, Transport Equations in Biology,, Birkhaeuser Verlag, (2007).

[54]

G. Rein, Non-linear stability of gaseous stars,, Arch. Rational Mech. Anal., 168 (2003), 115. doi: 10.1007/s00205-003-0260-y.

[55]

T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis,, Abstract and Applied Analysis, (2006). doi: 10.1155/AAA/2006/23061.

[56]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems,, Diff. Int. Equa., 19 (2006), 841.

[57]

Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models,, Adv. Diff. Eqns., 12 (2007), 121.

[58]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term,, J. Differential Equations, 227 (2006), 333. doi: 10.1016/j.jde.2006.03.003.

[59]

Y. Yao, Asymptotic behavior of radial solutions for critical Patlak-Keller-Segel model and an repulsive-attractive aggregation equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 81. doi: 10.1016/j.anihpc.2013.02.002.

show all references

References:
[1]

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.

[2]

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.

[3]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality,, Ann. Math., 138 (1993), 213. doi: 10.2307/2946638.

[4]

S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m\geq 0$,, Comm. Math. Phys., 323 (2013), 1017. doi: 10.1007/s00220-013-1777-z.

[5]

P. Biler and T. Nadzieja, A class of nonlocal parabolic problems occurring in statistical mechanics,, Colloq. Math., 66 (1993), 131.

[6]

A. Blanchet, E. A. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model,, J. Funct. Anal., 262 (2012), 2142. doi: 10.1016/j.jfa.2011.12.012.

[7]

A. Blanchet, On the Parabolic-Elliptic Patlak-Keller-Segel System in Dimension 2 and Higher,, Séminaire Laurent Schwartz$-$EDP et applications, (): 1.

[8]

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), 133. doi: 10.1007/s00526-008-0200-7.

[9]

A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbbR^2$,, Comm. Pure Appl. Math., 61 (2008), 1449. doi: 10.1002/cpa.20225.

[10]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,, Electron. J. Diff. Eqns., 44 (2006).

[11]

A. Blanchet and P. Laurençot, Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion,, Comm. Pure Appl. Math., 11 (2012), 47.

[12]

L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271. doi: 10.1002/cpa.3160420304.

[13]

E. Carlen, J. A. Carrillo and M. Loss, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows,, Proc. Nat. Acad. USA, 107 (2010), 19696. doi: 10.1073/pnas.1008323107.

[14]

E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $S^n$,, Geom. Funct. Anal., 2 (1992), 90. doi: 10.1007/BF01895706.

[15]

J. A. Carrillo, L. Chen, J.-G. Liu and J. Wang, A note on the subcritical two dimensional Keller-Segel system,, Acta appl. math., 119 (2012), 43. doi: 10.1007/s10440-011-9660-4.

[16]

L. Chen, J.-G. Liu and J. H. Wang, Multidimensional degenerate Keller-Segel system with critical diffusion exponent $2n/(n+2)$,, SIAM J. Math. Anal., 44 (2012), 1077. doi: 10.1137/110839102.

[17]

L. Chen and J. H. Wang, Exact criterion for global existence and blow-up to a degenerate Keller-Segel system,, Doc. Math., 19 (2014), 103.

[18]

X. Chen, A. Jüngel and J.-G. Liu, A note on Aubin-Lions-Dubinskii lemmas,, Acta Appl. Math., 133 (2014), 33. doi: 10.1007/s10440-013-9858-8.

[19]

W. X. Chen and C. M. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. Journal, 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8.

[20]

S. Childress, Chemotactic collapse in two dimensions,, Lect. Notes in Biomathematics, 55 (1984), 61. doi: 10.1007/978-3-642-45589-6_6.

[21]

T. Cieślak and P. Laurencot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchwski-Poisson system,, C. R. Acad. Sci. Paris Ser. I, 347 (2009), 237. doi: 10.1016/j.crma.2009.01.016.

[22]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, Nonlinearity, 21 (2008), 1057. doi: 10.1088/0951-7715/21/5/009.

[23]

J. M. Delort, Existence de nappes de tourbillon en dimension deux,, J. Amer. Math. Soc., 4 (1991), 553. doi: 10.1090/S0894-0347-1991-1102579-6.

[24]

R. DiPerna and A. Majda, Concentrations in regularizations for 2-D incompressible flow,, Comm. Pure Appl. Math., 40 (1987), 301. doi: 10.1002/cpa.3160400304.

[25]

J. Dolbeault, Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion,, Math. Res. Lett., 18 (2011), 1037. doi: 10.4310/MRL.2011.v18.n6.a1.

[26]

J. Dolbeault, M. J. Esteban and G. Jankowiak, The Moser-Trudinger-Onofri inequality, preprint,, , ().

[27]

J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional Keller-Segel model in $\mathbbR^2$,, C. R. Acad. Sci. Paris, 339 (2004), 611. doi: 10.1016/j.crma.2004.08.011.

[28]

R. L. Frank and E. H. Lieb, A new rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, Spectral theory, function spaces and inequalities,, Oper. Theory Adv. Appl., 219 (2012), 55. doi: 10.1007/978-3-0348-0263-5_4.

[29]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u + u^{1+\alpha}$,, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109.

[30]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77. doi: 10.1002/mana.19981950106.

[31]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure and Appl. Math., 34 (1981), 525. doi: 10.1002/cpa.3160340406.

[32]

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.

[33]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences I,, Jahresberichte der DMV, 105 (2003), 103.

[34]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences II,, Jahresberichte der DMV, 106 (2004), 51.

[35]

D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model,, J. Math. Biol., 44 (2002), 463. doi: 10.1007/s002850100134.

[36]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022.

[37]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, J. Differential Equations, 252 (2012), 1421. doi: 10.1016/j.jde.2011.02.012.

[38]

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. doi: 10.1090/S0002-9947-1992-1046835-6.

[39]

J. Jang, Nonlinear instability in gravitational Euler-Poisson systems for $\gamma=6/5$,, Arch. Rational Mech. Anal., 188 (2008), 265. doi: 10.1007/s00205-007-0086-0.

[40]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5.

[41]

E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theoret. Biol., 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6.

[42]

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. doi: 10.1137/110823584.

[43]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model,, J. Math. Anal. Appl., 343 (2008), 379. doi: 10.1016/j.jmaa.2008.01.005.

[44]

E. H. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities,, Ann. Math., 118 (1983), 349. doi: 10.2307/2007032.

[45]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14,, $2^{nd}$ edition, (2001). doi: 10.1090/gsm/014.

[46]

J.-G. Liu and Z. P. Xin, Convergence of Vortex Methods for Weak Solution to the 2-D Euler Equations with Vortex Sheet Data,, Comm. Pure Appl. Math., 48 (1995), 611. doi: 10.1002/cpa.3160480603.

[47]

J.-G. Liu and Z. P. Xin, Convergence of point vortex method for 2-D vortex sheet,, Math. Comp., 70 (2001), 595. doi: 10.1090/S0025-5718-00-01271-0.

[48]

S. Luckhaus and Y. Sugiyama, Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems,, M2AN Math. Model. Numer. Anal., 40 (2006), 597. doi: 10.1051/m2an:2006025.

[49]

A. J. Majda, Remarks on Weak Solution for Vortex Sheets with a Distinguished Sign,, Indiana Univ. Math. J., 42 (1993), 921. doi: 10.1512/iumj.1993.42.42043.

[50]

T. Nagai and T. Senba, Global existence and blow-up of radial solutioins to a parabolic-elliptic system of chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 145.

[51]

K. J. Painter and T. Hillen, Volume filling and quorum sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501.

[52]

C. S. Patlak, Random walk with persistenc and external bias,, Bull. Math. Biol. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407.

[53]

B. Perthame, Transport Equations in Biology,, Birkhaeuser Verlag, (2007).

[54]

G. Rein, Non-linear stability of gaseous stars,, Arch. Rational Mech. Anal., 168 (2003), 115. doi: 10.1007/s00205-003-0260-y.

[55]

T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis,, Abstract and Applied Analysis, (2006). doi: 10.1155/AAA/2006/23061.

[56]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems,, Diff. Int. Equa., 19 (2006), 841.

[57]

Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models,, Adv. Diff. Eqns., 12 (2007), 121.

[58]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term,, J. Differential Equations, 227 (2006), 333. doi: 10.1016/j.jde.2006.03.003.

[59]

Y. Yao, Asymptotic behavior of radial solutions for critical Patlak-Keller-Segel model and an repulsive-attractive aggregation equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 81. doi: 10.1016/j.anihpc.2013.02.002.

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