March  2016, 11(1): 181-201. doi: 10.3934/nhm.2016.11.181

On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion

1. 

Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo

2. 

Department of Mathematics, Faculty of Education, Zirve University, Gaziantep, 27260, Turkey

Received  April 2015 Revised  July 2015 Published  January 2016

We investigate a Keller-Segel model with quorum sensing and a fractional diffusion operator. This model describes the collective cell movement due to chemical sensing with flux limitation for high cell densities and with anomalous media represented by a nonlinear, degenerate fractional diffusion operator. The purpose of this paper is to introduce and prove the existence of a properly defined entropy solution.
Citation: Kenneth H. Karlsen, Süleyman Ulusoy. On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion. Networks & Heterogeneous Media, 2016, 11 (1) : 181-201. doi: 10.3934/nhm.2016.11.181
References:
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F. Bartumeus, F. Peters, S. Pueyo, C. Marrase and J. Katalan, Helical lévy walks: Adjusting searching statistics to resource availability in microzooplankton,, Proc. Natl. Acad. Sci., 100 (2003), 12771.  doi: 10.1073/pnas.2137243100.  Google Scholar

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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), 1683.  doi: 10.1088/0951-7715/24/6/001.  Google Scholar

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M. Bendahmane, K. H. Karlsen and J. M. Urbano, On a two-sidedly degenerate chemotaxis model with volume-filling effect,, Math. Methods Appl. Sci., 17 (2007), 783.  doi: 10.1142/S0218202507002108.  Google Scholar

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P. Biler, T. Funaki and W. A. Woyczynski, Fractal Burgers equations,, J. Differential Equations, 148 (1998), 9.  doi: 10.1006/jdeq.1998.3458.  Google Scholar

[7]

P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model,, J. Evol. Equ., 10 (2010), 247.  doi: 10.1007/s00028-009-0048-0.  Google Scholar

[8]

P. Biler, G. Karch and W. A. Woyczyński, Asymptotics for conservation laws involving Lévy diffusion generators,, Studia Math., 148 (2001), 171.  doi: 10.4064/sm148-2-5.  Google Scholar

[9]

P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems,, SIAM J. Appl. Math., 59 (1999), 845.  doi: 10.1137/S0036139996313447.  Google Scholar

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P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion,, Math. Methods Appl. Sci., 32 (2009), 112.  doi: 10.1002/mma.1036.  Google Scholar

[11]

A. Blanchet, J. A. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions,, Calc. Var. Partial Differential Equations, 35 (2009), 133.  doi: 10.1007/s00526-008-0200-7.  Google Scholar

[12]

A. Bueno-Orovio, D. Kay, V. Grau, B. Rodriguez and K. Burrage, Fractional diffusion models of cardiac electrical propagation: Role of structural heterogeneity in dispersion of repolarization,, J. R. Soc. Interface, 11 (2014).  doi: 10.1098/rsif.2014.0352.  Google Scholar

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N. Bournaveas and V. Calvez, The one-dimensional Keller-Segel model with fractional diffusion of cells,, Nonlinearity, 23 (2010), 923.  doi: 10.1088/0951-7715/23/4/009.  Google Scholar

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M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions,, Nonlinear Anal. Real World Appl., 8 (2007), 939.  doi: 10.1016/j.nonrwa.2006.04.002.  Google Scholar

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M. Burger, M. Di Francesco and Y. Dolak-Struss, The Keller-Segel model for chemotaxis with prevention of overcrowding: Linear vs. nonlinear diffusion,, SIAM J. Math. Anal., 38 (2006), 1288.  doi: 10.1137/050637923.  Google Scholar

[16]

M. Burger, Y. Dolak-Struss and C. Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions,, Commun. Math. Sci., 6 (2008), 1.  doi: 10.4310/CMS.2008.v6.n1.a1.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar

[18]

L. A. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation,, Arch. Ration. Mech. Anal., 195 (2010), 1.  doi: 10.1007/s00205-008-0181-x.  Google Scholar

[19]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann. of Math. (2), 171 (2010), 1903.  doi: 10.4007/annals.2010.171.1903.  Google Scholar

[20]

S. Cifani and E. R. Jakobsen, Entropy solution theory for fractional degenerate convection-diffusion equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 413.  doi: 10.1016/j.anihpc.2011.02.006.  Google Scholar

[21]

N. V. Chemetov, Nonlinear hyperbolic-elliptic systems in the bounded domain,, Commun. Pure Appl. Anal., 10 (2011), 1079.  doi: 10.3934/cpaa.2011.10.1079.  Google Scholar

[22]

N. Chemetov and W. Neves, The generalized Buckley-Leverett System: Solvability,, Arch. Ration. Mech. Anal., 208 (2013), 1.  doi: 10.1007/s00205-012-0591-7.  Google Scholar

[23]

G.-Q. Chen, Q. Ding and K. H. Karlsen, On nonlinear stochastic balance laws,, Arch. Ration. Mech. Anal., 204 (2012), 707.  doi: 10.1007/s00205-011-0489-9.  Google Scholar

[24]

G. M. Coclite, K. H. Karlsen, S. Mishra and N. H. Risebro, A hyperbolic-elliptic model of two-phase flow in porous media - existence of entropy solutions,, Int. J. Numer. Anal. Model., 9 (2012), 562.   Google Scholar

[25]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations,, Comm. Math. Phys., 249 (2004), 511.  doi: 10.1007/s00220-004-1055-1.  Google Scholar

[26]

A. Córdoba and D. Córdoba, A pointwise estimate for fractionary derivatives with applications to partial differential equations,, Proc. Natl. Acad. Sci. USA, 100 (2003), 15316.  doi: 10.1073/pnas.2036515100.  Google Scholar

[27]

A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing,, J. Funct. Anal., 259 (2010), 1014.  doi: 10.1016/j.jfa.2010.02.016.  Google Scholar

[28]

Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity,, SIAM J. Appl. Math., 66 (2005), 286.  doi: 10.1137/040612841.  Google Scholar

[29]

J. Droniou, T. Gallouet and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation,, J. Evol. Equ., 3 (2003), 499.  doi: 10.1007/s00028-003-0503-1.  Google Scholar

[30]

J. Droniou and C. Imbert, Fractal first-order partial differential equations,, Arch. Ration. Mech. Anal., 182 (2006), 299.  doi: 10.1007/s00205-006-0429-2.  Google Scholar

[31]

C. Escudero, Chemotactic collapse and mesenchymal morphogenesis,, Phys Rev E Stat Nonlin Soft Matter Phys, 72 (2005).  doi: 10.1103/PhysRevE.72.022903.  Google Scholar

[32]

C. Escudero, The fractional Keller-Segel model,, Nonlinearity, 19 (2006), 2909.  doi: 10.1088/0951-7715/19/12/010.  Google Scholar

[33]

L. C. Evans, Partial Differential Equations, volume 19 of {Graduate Studies in Mathematics,, American Mathematical Society, (2010).  doi: 10.1090/gsm/019.  Google Scholar

[34]

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

[35]

S. Jarohs and T. Weth, Asymptotic symmetry for a class of nonlinear fractional reaction diffusion equations,, Discrete and Continuous Dynamical Systems-A, 34 (2014), 2581.  doi: 10.3934/dcds.2014.34.2581.  Google Scholar

[36]

K. H. Karlsen and S. Ulusoy, Stability of entropy solutions for Lévy mixed hyperbolic-parabolic equations,, Electron. J. Differential Equations, 2011 (2011), 1.   Google Scholar

[37]

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

[38]

E. F. Keller and L. A. Segel, Model for chemotaxis,, Journal of Theoretical Biology, 30 (1971), 225.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[39]

A. Kiselev, F. Nazarov and R. Shterenberg, Blow up and regularity for fractal Burgers equation,, Dyn. Partial Differ. Equ., 5 (2008), 211.  doi: 10.4310/DPDE.2008.v5.n3.a2.  Google Scholar

[40]

J. Klafter, B. S. White and M. Levandowsky, Microzooplankton feeding behavior and the Lévy walks,, Biological Motion, 89 (1990), 281.  doi: 10.1007/978-3-642-51664-1_20.  Google Scholar

[41]

S. N. Kružkov, Results on the nature of the continuity of solutions of parabolic equations, and certain applications thereof,, Mat. Zametki, 6 (1969), 97.   Google Scholar

[42]

S. N. Kružkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81 (1970), 228.   Google Scholar

[43]

M. Levandowsky, B. S. White and F. L. Schuster, Random movements of soil amebas,, Acta Protozool, 36 (1997), 237.   Google Scholar

[44]

F. Matthäus, M. S. Mommer, T. Curk and J. Dobnikar, On the origin and characteristics of noise-induced Lévy walks of E. Coli,, PLoS ONE, 6 (2011).   Google Scholar

[45]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach,, Phys. Rep., 339 (2000), 1.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[46]

F. Otto, $L^1-$contraction and uniqueness for quasilinear elliptic-parabolic equations,, C. R. Acad. Sci. Paris Sér I Math., 321 (1995), 1005.  doi: 10.1006/jdeq.1996.0155.  Google Scholar

[47]

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

[48]

B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model,, Trans. Amer. Math. Soc., 361 (2009), 2319.  doi: 10.1090/S0002-9947-08-04656-4.  Google Scholar

[49]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196.  doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[50]

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

[51]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bull. Math. Biol., 68 (2006), 1601.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[52]

G. Wu and X. Zheng, On the well-posedness for Keller-Segel system with fractional diffusion,, Math. Methods Appl. Sci., 34 (2011), 1739.  doi: 10.1002/mma.1480.  Google Scholar

show all references

References:
[1]

N. Alibaud, Entropy formulation for fractal conservation laws,, J. Evol. Equ., 7 (2007), 145.  doi: 10.1007/s00028-006-0253-z.  Google Scholar

[2]

F. Bartumeus, F. Peters, S. Pueyo, C. Marrase and J. Katalan, Helical lévy walks: Adjusting searching statistics to resource availability in microzooplankton,, Proc. Natl. Acad. Sci., 100 (2003), 12771.  doi: 10.1073/pnas.2137243100.  Google Scholar

[3]

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), 1683.  doi: 10.1088/0951-7715/24/6/001.  Google Scholar

[4]

F. Ben Belgacem and P.-E. Jabin, Compactness for nonlinear transport equations,, J. Funct. Anal., 264 (2013), 139.  doi: 10.1016/j.jfa.2012.10.005.  Google Scholar

[5]

M. Bendahmane, K. H. Karlsen and J. M. Urbano, On a two-sidedly degenerate chemotaxis model with volume-filling effect,, Math. Methods Appl. Sci., 17 (2007), 783.  doi: 10.1142/S0218202507002108.  Google Scholar

[6]

P. Biler, T. Funaki and W. A. Woyczynski, Fractal Burgers equations,, J. Differential Equations, 148 (1998), 9.  doi: 10.1006/jdeq.1998.3458.  Google Scholar

[7]

P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model,, J. Evol. Equ., 10 (2010), 247.  doi: 10.1007/s00028-009-0048-0.  Google Scholar

[8]

P. Biler, G. Karch and W. A. Woyczyński, Asymptotics for conservation laws involving Lévy diffusion generators,, Studia Math., 148 (2001), 171.  doi: 10.4064/sm148-2-5.  Google Scholar

[9]

P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems,, SIAM J. Appl. Math., 59 (1999), 845.  doi: 10.1137/S0036139996313447.  Google Scholar

[10]

P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion,, Math. Methods Appl. Sci., 32 (2009), 112.  doi: 10.1002/mma.1036.  Google Scholar

[11]

A. Blanchet, J. A. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions,, Calc. Var. Partial Differential Equations, 35 (2009), 133.  doi: 10.1007/s00526-008-0200-7.  Google Scholar

[12]

A. Bueno-Orovio, D. Kay, V. Grau, B. Rodriguez and K. Burrage, Fractional diffusion models of cardiac electrical propagation: Role of structural heterogeneity in dispersion of repolarization,, J. R. Soc. Interface, 11 (2014).  doi: 10.1098/rsif.2014.0352.  Google Scholar

[13]

N. Bournaveas and V. Calvez, The one-dimensional Keller-Segel model with fractional diffusion of cells,, Nonlinearity, 23 (2010), 923.  doi: 10.1088/0951-7715/23/4/009.  Google Scholar

[14]

M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions,, Nonlinear Anal. Real World Appl., 8 (2007), 939.  doi: 10.1016/j.nonrwa.2006.04.002.  Google Scholar

[15]

M. Burger, M. Di Francesco and Y. Dolak-Struss, The Keller-Segel model for chemotaxis with prevention of overcrowding: Linear vs. nonlinear diffusion,, SIAM J. Math. Anal., 38 (2006), 1288.  doi: 10.1137/050637923.  Google Scholar

[16]

M. Burger, Y. Dolak-Struss and C. Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions,, Commun. Math. Sci., 6 (2008), 1.  doi: 10.4310/CMS.2008.v6.n1.a1.  Google Scholar

[17]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar

[18]

L. A. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation,, Arch. Ration. Mech. Anal., 195 (2010), 1.  doi: 10.1007/s00205-008-0181-x.  Google Scholar

[19]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann. of Math. (2), 171 (2010), 1903.  doi: 10.4007/annals.2010.171.1903.  Google Scholar

[20]

S. Cifani and E. R. Jakobsen, Entropy solution theory for fractional degenerate convection-diffusion equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 413.  doi: 10.1016/j.anihpc.2011.02.006.  Google Scholar

[21]

N. V. Chemetov, Nonlinear hyperbolic-elliptic systems in the bounded domain,, Commun. Pure Appl. Anal., 10 (2011), 1079.  doi: 10.3934/cpaa.2011.10.1079.  Google Scholar

[22]

N. Chemetov and W. Neves, The generalized Buckley-Leverett System: Solvability,, Arch. Ration. Mech. Anal., 208 (2013), 1.  doi: 10.1007/s00205-012-0591-7.  Google Scholar

[23]

G.-Q. Chen, Q. Ding and K. H. Karlsen, On nonlinear stochastic balance laws,, Arch. Ration. Mech. Anal., 204 (2012), 707.  doi: 10.1007/s00205-011-0489-9.  Google Scholar

[24]

G. M. Coclite, K. H. Karlsen, S. Mishra and N. H. Risebro, A hyperbolic-elliptic model of two-phase flow in porous media - existence of entropy solutions,, Int. J. Numer. Anal. Model., 9 (2012), 562.   Google Scholar

[25]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations,, Comm. Math. Phys., 249 (2004), 511.  doi: 10.1007/s00220-004-1055-1.  Google Scholar

[26]

A. Córdoba and D. Córdoba, A pointwise estimate for fractionary derivatives with applications to partial differential equations,, Proc. Natl. Acad. Sci. USA, 100 (2003), 15316.  doi: 10.1073/pnas.2036515100.  Google Scholar

[27]

A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing,, J. Funct. Anal., 259 (2010), 1014.  doi: 10.1016/j.jfa.2010.02.016.  Google Scholar

[28]

Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity,, SIAM J. Appl. Math., 66 (2005), 286.  doi: 10.1137/040612841.  Google Scholar

[29]

J. Droniou, T. Gallouet and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation,, J. Evol. Equ., 3 (2003), 499.  doi: 10.1007/s00028-003-0503-1.  Google Scholar

[30]

J. Droniou and C. Imbert, Fractal first-order partial differential equations,, Arch. Ration. Mech. Anal., 182 (2006), 299.  doi: 10.1007/s00205-006-0429-2.  Google Scholar

[31]

C. Escudero, Chemotactic collapse and mesenchymal morphogenesis,, Phys Rev E Stat Nonlin Soft Matter Phys, 72 (2005).  doi: 10.1103/PhysRevE.72.022903.  Google Scholar

[32]

C. Escudero, The fractional Keller-Segel model,, Nonlinearity, 19 (2006), 2909.  doi: 10.1088/0951-7715/19/12/010.  Google Scholar

[33]

L. C. Evans, Partial Differential Equations, volume 19 of {Graduate Studies in Mathematics,, American Mathematical Society, (2010).  doi: 10.1090/gsm/019.  Google Scholar

[34]

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

[35]

S. Jarohs and T. Weth, Asymptotic symmetry for a class of nonlinear fractional reaction diffusion equations,, Discrete and Continuous Dynamical Systems-A, 34 (2014), 2581.  doi: 10.3934/dcds.2014.34.2581.  Google Scholar

[36]

K. H. Karlsen and S. Ulusoy, Stability of entropy solutions for Lévy mixed hyperbolic-parabolic equations,, Electron. J. Differential Equations, 2011 (2011), 1.   Google Scholar

[37]

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

[38]

E. F. Keller and L. A. Segel, Model for chemotaxis,, Journal of Theoretical Biology, 30 (1971), 225.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[39]

A. Kiselev, F. Nazarov and R. Shterenberg, Blow up and regularity for fractal Burgers equation,, Dyn. Partial Differ. Equ., 5 (2008), 211.  doi: 10.4310/DPDE.2008.v5.n3.a2.  Google Scholar

[40]

J. Klafter, B. S. White and M. Levandowsky, Microzooplankton feeding behavior and the Lévy walks,, Biological Motion, 89 (1990), 281.  doi: 10.1007/978-3-642-51664-1_20.  Google Scholar

[41]

S. N. Kružkov, Results on the nature of the continuity of solutions of parabolic equations, and certain applications thereof,, Mat. Zametki, 6 (1969), 97.   Google Scholar

[42]

S. N. Kružkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81 (1970), 228.   Google Scholar

[43]

M. Levandowsky, B. S. White and F. L. Schuster, Random movements of soil amebas,, Acta Protozool, 36 (1997), 237.   Google Scholar

[44]

F. Matthäus, M. S. Mommer, T. Curk and J. Dobnikar, On the origin and characteristics of noise-induced Lévy walks of E. Coli,, PLoS ONE, 6 (2011).   Google Scholar

[45]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach,, Phys. Rep., 339 (2000), 1.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[46]

F. Otto, $L^1-$contraction and uniqueness for quasilinear elliptic-parabolic equations,, C. R. Acad. Sci. Paris Sér I Math., 321 (1995), 1005.  doi: 10.1006/jdeq.1996.0155.  Google Scholar

[47]

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

[48]

B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model,, Trans. Amer. Math. Soc., 361 (2009), 2319.  doi: 10.1090/S0002-9947-08-04656-4.  Google Scholar

[49]

J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196.  doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[50]

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

[51]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bull. Math. Biol., 68 (2006), 1601.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[52]

G. Wu and X. Zheng, On the well-posedness for Keller-Segel system with fractional diffusion,, Math. Methods Appl. Sci., 34 (2011), 1739.  doi: 10.1002/mma.1480.  Google Scholar

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