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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. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[5]

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

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[44]

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

[45]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[51]

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

[52]

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

[53]

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

[54]

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

[55]

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

[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. Google Scholar

[57]

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

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[5]

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

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[44]

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

[45]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[51]

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

[52]

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

[53]

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

[54]

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

[55]

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

[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. Google Scholar

[57]

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

[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. Google Scholar

[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. Google Scholar

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