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 and Heterogeneous Media, 2016, 11 (1) : 181-201. doi: 10.3934/nhm.2016.11.181
References:
[1]

N. Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ., 7 (2007), 145-175. doi: 10.1007/s00028-006-0253-z.

[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), 12771-12775. doi: 10.1073/pnas.2137243100.

[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), 1683-1714. doi: 10.1088/0951-7715/24/6/001.

[4]

F. Ben Belgacem and P.-E. Jabin, Compactness for nonlinear transport equations, J. Funct. Anal., 264 (2013), 139-168. doi: 10.1016/j.jfa.2012.10.005.

[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), 783-804. doi: 10.1142/S0218202507002108.

[6]

P. Biler, T. Funaki and W. A. Woyczynski, Fractal Burgers equations, J. Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458.

[7]

P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262. doi: 10.1007/s00028-009-0048-0.

[8]

P. Biler, G. Karch and W. A. Woyczyński, Asymptotics for conservation laws involving Lévy diffusion generators, Studia Math., 148 (2001), 171-192. doi: 10.4064/sm148-2-5.

[9]

P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869 (electronic). doi: 10.1137/S0036139996313447.

[10]

P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Methods Appl. Sci., 32 (2009), 112-126. doi: 10.1002/mma.1036.

[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), 133-168. doi: 10.1007/s00526-008-0200-7.

[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), 20140352. doi: 10.1098/rsif.2014.0352.

[13]

N. Bournaveas and V. Calvez, The one-dimensional Keller-Segel model with fractional diffusion of cells, Nonlinearity, 23 (2010), 923-935. doi: 10.1088/0951-7715/23/4/009.

[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), 939-958. doi: 10.1016/j.nonrwa.2006.04.002.

[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), 1288-1315 (electronic). doi: 10.1137/050637923.

[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), 1-28. doi: 10.4310/CMS.2008.v6.n1.a1.

[17]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[18]

L. A. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal., 195 (2010), 1-23. doi: 10.1007/s00205-008-0181-x.

[19]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.

[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), 413-441. doi: 10.1016/j.anihpc.2011.02.006.

[21]

N. V. Chemetov, Nonlinear hyperbolic-elliptic systems in the bounded domain, Commun. Pure Appl. Anal., 10 (2011), 1079-1096. doi: 10.3934/cpaa.2011.10.1079.

[22]

N. Chemetov and W. Neves, The generalized Buckley-Leverett System: Solvability, Arch. Ration. Mech. Anal., 208 (2013), 1-24. doi: 10.1007/s00205-012-0591-7.

[23]

G.-Q. Chen, Q. Ding and K. H. Karlsen, On nonlinear stochastic balance laws, Arch. Ration. Mech. Anal., 204 (2012), 707-743. doi: 10.1007/s00205-011-0489-9.

[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), 562-583.

[25]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.

[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), 15316-15317. doi: 10.1073/pnas.2036515100.

[27]

A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing, J. Funct. Anal., 259 (2010), 1014-1042. doi: 10.1016/j.jfa.2010.02.016.

[28]

Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math., 66 (2005), 286-308 (electronic). doi: 10.1137/040612841.

[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), 499-521. doi: 10.1007/s00028-003-0503-1.

[30]

J. Droniou and C. Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331. doi: 10.1007/s00205-006-0429-2.

[31]

C. Escudero, Chemotactic collapse and mesenchymal morphogenesis, Phys Rev E Stat Nonlin Soft Matter Phys, 72 (2005), 022903. doi: 10.1103/PhysRevE.72.022903.

[32]

C. Escudero, The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918. doi: 10.1088/0951-7715/19/12/010.

[33]

L. C. Evans, Partial Differential Equations, volume 19 of {Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010. doi: 10.1090/gsm/019.

[34]

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.

[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), 2581-2615. doi: 10.3934/dcds.2014.34.2581.

[36]

K. H. Karlsen and S. Ulusoy, Stability of entropy solutions for Lévy mixed hyperbolic-parabolic equations, Electron. J. Differential Equations, 2011 (2011), 1-23.

[37]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[38]

E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.

[39]

A. Kiselev, F. Nazarov and R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ., 5 (2008), 211-240. doi: 10.4310/DPDE.2008.v5.n3.a2.

[40]

J. Klafter, B. S. White and M. Levandowsky, Microzooplankton feeding behavior and the Lévy walks, Biological Motion, Lecture notes in Biomathematics (ed. W. Alt and G. Hoffmann), Springer, 89 (1990), 281-296. doi: 10.1007/978-3-642-51664-1_20.

[41]

S. N. Kružkov, Results on the nature of the continuity of solutions of parabolic equations, and certain applications thereof, Mat. Zametki, 6 (1969), 97-108.

[42]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.

[43]

M. Levandowsky, B. S. White and F. L. Schuster, Random movements of soil amebas, Acta Protozool, 36 (1997), 237-248.

[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), e18623.

[45]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[46]

F. Otto, $L^1-$contraction and uniqueness for quasilinear elliptic-parabolic equations, C. R. Acad. Sci. Paris Sér I Math., 321 (1995), 1005-1010. doi: 10.1006/jdeq.1996.0155.

[47]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.

[48]

B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335. doi: 10.1090/S0002-9947-08-04656-4.

[49]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.

[50]

Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144.

[51]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6.

[52]

G. Wu and X. Zheng, On the well-posedness for Keller-Segel system with fractional diffusion, Math. Methods Appl. Sci., 34 (2011), 1739-1750. doi: 10.1002/mma.1480.

show all references

References:
[1]

N. Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ., 7 (2007), 145-175. doi: 10.1007/s00028-006-0253-z.

[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), 12771-12775. doi: 10.1073/pnas.2137243100.

[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), 1683-1714. doi: 10.1088/0951-7715/24/6/001.

[4]

F. Ben Belgacem and P.-E. Jabin, Compactness for nonlinear transport equations, J. Funct. Anal., 264 (2013), 139-168. doi: 10.1016/j.jfa.2012.10.005.

[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), 783-804. doi: 10.1142/S0218202507002108.

[6]

P. Biler, T. Funaki and W. A. Woyczynski, Fractal Burgers equations, J. Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458.

[7]

P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262. doi: 10.1007/s00028-009-0048-0.

[8]

P. Biler, G. Karch and W. A. Woyczyński, Asymptotics for conservation laws involving Lévy diffusion generators, Studia Math., 148 (2001), 171-192. doi: 10.4064/sm148-2-5.

[9]

P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869 (electronic). doi: 10.1137/S0036139996313447.

[10]

P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Methods Appl. Sci., 32 (2009), 112-126. doi: 10.1002/mma.1036.

[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), 133-168. doi: 10.1007/s00526-008-0200-7.

[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), 20140352. doi: 10.1098/rsif.2014.0352.

[13]

N. Bournaveas and V. Calvez, The one-dimensional Keller-Segel model with fractional diffusion of cells, Nonlinearity, 23 (2010), 923-935. doi: 10.1088/0951-7715/23/4/009.

[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), 939-958. doi: 10.1016/j.nonrwa.2006.04.002.

[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), 1288-1315 (electronic). doi: 10.1137/050637923.

[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), 1-28. doi: 10.4310/CMS.2008.v6.n1.a1.

[17]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[18]

L. A. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal., 195 (2010), 1-23. doi: 10.1007/s00205-008-0181-x.

[19]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.

[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), 413-441. doi: 10.1016/j.anihpc.2011.02.006.

[21]

N. V. Chemetov, Nonlinear hyperbolic-elliptic systems in the bounded domain, Commun. Pure Appl. Anal., 10 (2011), 1079-1096. doi: 10.3934/cpaa.2011.10.1079.

[22]

N. Chemetov and W. Neves, The generalized Buckley-Leverett System: Solvability, Arch. Ration. Mech. Anal., 208 (2013), 1-24. doi: 10.1007/s00205-012-0591-7.

[23]

G.-Q. Chen, Q. Ding and K. H. Karlsen, On nonlinear stochastic balance laws, Arch. Ration. Mech. Anal., 204 (2012), 707-743. doi: 10.1007/s00205-011-0489-9.

[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), 562-583.

[25]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.

[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), 15316-15317. doi: 10.1073/pnas.2036515100.

[27]

A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing, J. Funct. Anal., 259 (2010), 1014-1042. doi: 10.1016/j.jfa.2010.02.016.

[28]

Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math., 66 (2005), 286-308 (electronic). doi: 10.1137/040612841.

[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), 499-521. doi: 10.1007/s00028-003-0503-1.

[30]

J. Droniou and C. Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331. doi: 10.1007/s00205-006-0429-2.

[31]

C. Escudero, Chemotactic collapse and mesenchymal morphogenesis, Phys Rev E Stat Nonlin Soft Matter Phys, 72 (2005), 022903. doi: 10.1103/PhysRevE.72.022903.

[32]

C. Escudero, The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918. doi: 10.1088/0951-7715/19/12/010.

[33]

L. C. Evans, Partial Differential Equations, volume 19 of {Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010. doi: 10.1090/gsm/019.

[34]

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.

[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), 2581-2615. doi: 10.3934/dcds.2014.34.2581.

[36]

K. H. Karlsen and S. Ulusoy, Stability of entropy solutions for Lévy mixed hyperbolic-parabolic equations, Electron. J. Differential Equations, 2011 (2011), 1-23.

[37]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.

[38]

E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.

[39]

A. Kiselev, F. Nazarov and R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ., 5 (2008), 211-240. doi: 10.4310/DPDE.2008.v5.n3.a2.

[40]

J. Klafter, B. S. White and M. Levandowsky, Microzooplankton feeding behavior and the Lévy walks, Biological Motion, Lecture notes in Biomathematics (ed. W. Alt and G. Hoffmann), Springer, 89 (1990), 281-296. doi: 10.1007/978-3-642-51664-1_20.

[41]

S. N. Kružkov, Results on the nature of the continuity of solutions of parabolic equations, and certain applications thereof, Mat. Zametki, 6 (1969), 97-108.

[42]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.

[43]

M. Levandowsky, B. S. White and F. L. Schuster, Random movements of soil amebas, Acta Protozool, 36 (1997), 237-248.

[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), e18623.

[45]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[46]

F. Otto, $L^1-$contraction and uniqueness for quasilinear elliptic-parabolic equations, C. R. Acad. Sci. Paris Sér I Math., 321 (1995), 1005-1010. doi: 10.1006/jdeq.1996.0155.

[47]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.

[48]

B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335. doi: 10.1090/S0002-9947-08-04656-4.

[49]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.

[50]

Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144.

[51]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6.

[52]

G. Wu and X. Zheng, On the well-posedness for Keller-Segel system with fractional diffusion, Math. Methods Appl. Sci., 34 (2011), 1739-1750. doi: 10.1002/mma.1480.

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