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November  2014, 34(11): 4911-4946. doi: 10.3934/dcds.2014.34.4911

On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model

Tian Xiang 1,

1. 

Department of Mathematics, Tulane University, New Orleans, LA 70118

Received  September 2013 Revised  February 2014 Published  May 2014

In this paper, we continue to study a general nonlocal gradient Patlak-Keller-Segel chemotaxis model in a one dimensional spatial domain. By utilizing the properties of the nonlocal gradient, we first apply the well-known Moser-Alikakos iteration technique plus the heat semigroup theory to obtain the boundedness and hence the global existence of its solution. Then we study the asymptotic behavior of the time-dependent solution, and obtain the limiting equations when the sampling radius $\rho\rightarrow 0$ as well as convergence results when time $t\rightarrow \infty$. Along this way, a ``global" stability issue of the spiky stationary solution for the minimal model is formulated. Finally and importantly, we study the stability of the nonconstant bifurcating solutions. Interestingly, the small size of the cells enhances the occurrence of pattern formation, the stability results are independent of the net creation rate of the chemical, and the stability is closely related to the cell radius $\rho$. Typically, when the cell (net) degradation rate lies below a threshold (stabilizing) value, the cell is stable. Surprisingly, this threshold value is an increasing function of the cell radius. The large cells can compensate their degradation of the chemical signal, and become stable; however, for small cells to be stable, their degradation rate must be less than a threshold value.
Keywords: Chemotaxis, stability., boundedness, nonlocal gradient, bifurcation, asymptotic behavior.
Mathematics Subject Classification: Primary: 35K57, 35B40, 92C17, 35B32; Secondary: 35K45, 35J67, 35B41, 35B3.
Citation: Tian Xiang. On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4911-4946. doi: 10.3934/dcds.2014.34.4911
References:
[1]

R. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar

[2]

N. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827. doi: 10.1080/03605307908820113. Google Scholar

[3]

H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13. Google Scholar

[4]

H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence,, Math. Z., 202 (1989), 219. doi: 10.1007/BF01215256. 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,, Electron. J. Differential Equations 2006, (2006). Google Scholar

[6]

A. Blanchet, J. 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

[7]

A. Blanchet, J. Carrillo and P. Laurencot, 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

[8]

A. Blanchet, E. Carlen and J. 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

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A. Blanchet, On the Parabolic-Elliptic Patlak-Keller-Segel System in Dimension 2 and Higher,, preprint, (). Google Scholar

[10]

J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. blowup in a fully parabolic quasilinear 1D Keller-Segel system,, Nonlinear Anal., 75 (2012), 5215. doi: 10.1016/j.na.2012.04.038. Google Scholar

[11]

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

[12]

J. Campos and J. Dolbeault, Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane,, Communications in Partial Differential Equations, 39 (2014), 806. doi: 10.1080/03605302.2014.885046. Google Scholar

[13]

X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, Stability of spiky solution of the Keller-Segel's minimal chemotaxis model,, in process., (). Google Scholar

[14]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a Chemotaxis Model with Saturated Chemotactic Flux,, Kinetic and Related Models, 5 (2012), 51. doi: 10.3934/krm.2012.5.51. Google Scholar

[15]

S. Childress and J. Percus, Nonlinear aspects of chemotaxis,, Math. Biosci., 56 (1981), 217. doi: 10.1016/0025-5564(81)90055-9. Google Scholar

[16]

T. Cieślak and P. Laurencot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system,, Ann. Inst. H. Poincaré Anal. Non Linéire, 27 (2010), 437. doi: 10.1016/j.anihpc.2009.11.016. Google Scholar

[17]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, J. Differential Equations, 252 (2012), 5832. doi: 10.1016/j.jde.2012.01.045. Google Scholar

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[19]

M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161. Google Scholar

[20]

E. Feireisl, P. Laurencot and H. Petzeltova, On convergence to equilibria for the Keller-Segel chemotaxis model,, J. Differential Equations, 236 (2007), 551. doi: 10.1016/j.jde.2007.02.002. Google Scholar

[21]

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

[22]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar

[23]

T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile,, Math. Methods Appl. Sci., 27 (2004), 1783. doi: 10.1002/mma.569. Google Scholar

[24]

T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius,, Discrete Coin. Dyn. Syst. Ser. B, 7 (2007), 125. doi: 10.3934/dcdsb.2007.7.125. Google Scholar

[25]

T. Hillen and K. 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

[26]

D. Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: Some recent results,, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 399. doi: 10.1007/PL00001455. Google Scholar

[27]

D. Horstmann, Lyapunov functions and $L^p$-estimates for a class of reaction-diffusion systems,, Colloq. Math., 87 (2001), 113. doi: 10.4064/cm87-1-7. Google Scholar

[28]

D. Horstmann, From 1970 until now: The Keller-Segal model in chaemotaxis and its consequence I,, Jahresber DMV, 105 (2003), 103. Google Scholar

[29]

D. Horstmann, From 1970 until now: The Keller-Segal model in chaemotaxis and its consequence II,, Jahresber DMV, 106 (2004), 51. Google Scholar

[30]

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

[31]

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

[32]

J. Jiang and Y. Zhang, On convergence to equilibria for a chemotaxis model with volume-filling effect,, Asymptot. Anal., 65 (2009), 79. Google Scholar

[33]

K. Kang, T. Kolokolnikov and M. Ward, The stability and dynamics of a spike in the 1D Keller-Segel model,, IMA J. Appl. Math., 72 (2007), 140. doi: 10.1093/imamat/hxl028. Google Scholar

[34]

E. Keller and L. 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

[35]

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

[36]

O. Ladyzenskaja, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type,, AMS, (1968). Google Scholar

[37]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morpholog,, J. Theor. Biol., 42 (1973), 63. doi: 10.1016/0022-5193(73)90149-5. Google Scholar

[38]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441. Google Scholar

[39]

H. Othmer and T. Hillen, The diffusion limit of transport equations. II. Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772. Google Scholar

[40]

B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539. doi: 10.1007/s10492-004-6431-9. Google Scholar

[41]

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

[42]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Funct. Anal., 7 (1971), 487. doi: 10.1016/0022-1236(71)90030-9. Google Scholar

[43]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788. doi: 10.1016/j.jde.2008.09.009. Google Scholar

[44]

B. Sleeman, M. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model,, SIAM J. Appl. Math., 65 (2005), 790. doi: 10.1137/S0036139902415117. Google Scholar

[45]

J. Velazquez, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions,, SIAM J. Appl. Math., 64 (2004), 1198. doi: 10.1137/S0036139903433888. Google Scholar

[46]

X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics,, SIAM J. Math. Anal., 31 (2000), 535. doi: 10.1137/S0036141098339897. Google Scholar

[47]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation method and Helly's compactness theorem,, J. Math. Biol., 66 (2013), 1241. doi: 10.1007/s00285-012-0533-x. Google Scholar

[48]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748. doi: 10.1016/j.matpur.2013.01.020. Google Scholar

[49]

T. Xiang, A study on the positive nonconstant steady states of nonlocal chemotaxis systems,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2457. doi: 10.3934/dcdsb.2013.18.2457. Google Scholar

[50]

Y. Zhang, The steady states and convergence to equilibria for a 1-D chemotaxis model with volume-filling effect,, Math. Methods Appl. Sci., 33 (2010), 25. doi: 10.1002/mma.1283. Google Scholar

show all references

References:
[1]

R. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar

[2]

N. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827. doi: 10.1080/03605307908820113. Google Scholar

[3]

H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13. Google Scholar

[4]

H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence,, Math. Z., 202 (1989), 219. doi: 10.1007/BF01215256. Google Scholar

[5]

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

[6]

A. Blanchet, J. 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

[7]

A. Blanchet, J. Carrillo and P. Laurencot, 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

[8]

A. Blanchet, E. Carlen and J. 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

[9]

A. Blanchet, On the Parabolic-Elliptic Patlak-Keller-Segel System in Dimension 2 and Higher,, preprint, (). Google Scholar

[10]

J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. blowup in a fully parabolic quasilinear 1D Keller-Segel system,, Nonlinear Anal., 75 (2012), 5215. doi: 10.1016/j.na.2012.04.038. Google Scholar

[11]

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

[12]

J. Campos and J. Dolbeault, Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane,, Communications in Partial Differential Equations, 39 (2014), 806. doi: 10.1080/03605302.2014.885046. Google Scholar

[13]

X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, Stability of spiky solution of the Keller-Segel's minimal chemotaxis model,, in process., (). Google Scholar

[14]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a Chemotaxis Model with Saturated Chemotactic Flux,, Kinetic and Related Models, 5 (2012), 51. doi: 10.3934/krm.2012.5.51. Google Scholar

[15]

S. Childress and J. Percus, Nonlinear aspects of chemotaxis,, Math. Biosci., 56 (1981), 217. doi: 10.1016/0025-5564(81)90055-9. Google Scholar

[16]

T. Cieślak and P. Laurencot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system,, Ann. Inst. H. Poincaré Anal. Non Linéire, 27 (2010), 437. doi: 10.1016/j.anihpc.2009.11.016. Google Scholar

[17]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, J. Differential Equations, 252 (2012), 5832. doi: 10.1016/j.jde.2012.01.045. Google Scholar

[18]

M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321. Google Scholar

[19]

M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161. Google Scholar

[20]

E. Feireisl, P. Laurencot and H. Petzeltova, On convergence to equilibria for the Keller-Segel chemotaxis model,, J. Differential Equations, 236 (2007), 551. doi: 10.1016/j.jde.2007.02.002. Google Scholar

[21]

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

[22]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar

[23]

T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile,, Math. Methods Appl. Sci., 27 (2004), 1783. doi: 10.1002/mma.569. Google Scholar

[24]

T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius,, Discrete Coin. Dyn. Syst. Ser. B, 7 (2007), 125. doi: 10.3934/dcdsb.2007.7.125. Google Scholar

[25]

T. Hillen and K. 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

[26]

D. Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: Some recent results,, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 399. doi: 10.1007/PL00001455. Google Scholar

[27]

D. Horstmann, Lyapunov functions and $L^p$-estimates for a class of reaction-diffusion systems,, Colloq. Math., 87 (2001), 113. doi: 10.4064/cm87-1-7. Google Scholar

[28]

D. Horstmann, From 1970 until now: The Keller-Segal model in chaemotaxis and its consequence I,, Jahresber DMV, 105 (2003), 103. Google Scholar

[29]

D. Horstmann, From 1970 until now: The Keller-Segal model in chaemotaxis and its consequence II,, Jahresber DMV, 106 (2004), 51. Google Scholar

[30]

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

[31]

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

[32]

J. Jiang and Y. Zhang, On convergence to equilibria for a chemotaxis model with volume-filling effect,, Asymptot. Anal., 65 (2009), 79. Google Scholar

[33]

K. Kang, T. Kolokolnikov and M. Ward, The stability and dynamics of a spike in the 1D Keller-Segel model,, IMA J. Appl. Math., 72 (2007), 140. doi: 10.1093/imamat/hxl028. Google Scholar

[34]

E. Keller and L. 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

[35]

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

[36]

O. Ladyzenskaja, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type,, AMS, (1968). Google Scholar

[37]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morpholog,, J. Theor. Biol., 42 (1973), 63. doi: 10.1016/0022-5193(73)90149-5. Google Scholar

[38]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441. Google Scholar

[39]

H. Othmer and T. Hillen, The diffusion limit of transport equations. II. Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772. Google Scholar

[40]

B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539. doi: 10.1007/s10492-004-6431-9. Google Scholar

[41]

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

[42]

P. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Funct. Anal., 7 (1971), 487. doi: 10.1016/0022-1236(71)90030-9. Google Scholar

[43]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788. doi: 10.1016/j.jde.2008.09.009. Google Scholar

[44]

B. Sleeman, M. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model,, SIAM J. Appl. Math., 65 (2005), 790. doi: 10.1137/S0036139902415117. Google Scholar

[45]

J. Velazquez, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions,, SIAM J. Appl. Math., 64 (2004), 1198. doi: 10.1137/S0036139903433888. Google Scholar

[46]

X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics,, SIAM J. Math. Anal., 31 (2000), 535. doi: 10.1137/S0036141098339897. Google Scholar

[47]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation method and Helly's compactness theorem,, J. Math. Biol., 66 (2013), 1241. doi: 10.1007/s00285-012-0533-x. Google Scholar

[48]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748. doi: 10.1016/j.matpur.2013.01.020. Google Scholar

[49]

T. Xiang, A study on the positive nonconstant steady states of nonlocal chemotaxis systems,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2457. doi: 10.3934/dcdsb.2013.18.2457. Google Scholar

[50]

Y. Zhang, The steady states and convergence to equilibria for a 1-D chemotaxis model with volume-filling effect,, Math. Methods Appl. Sci., 33 (2010), 25. doi: 10.1002/mma.1283. Google Scholar

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