# American Institute of Mathematical Sciences

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

 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.
Citation: Tian Xiang. On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4911-4946. doi: 10.3934/dcds.2014.34.4911
##### References:
 [1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] N. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113. [3] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. [4] H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence, Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256. [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, No. 44, 32 pp. [6] A. Blanchet, J. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbb{R}^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481. doi: 10.1002/cpa.20225. [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), 133-168. doi: 10.1007/s00526-008-0200-7. [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), 2142-2230. doi: 10.1016/j.jfa.2011.12.012. [9] A. Blanchet, On the Parabolic-Elliptic Patlak-Keller-Segel System in Dimension 2 and Higher, preprint, arXiv:1109.1543. [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), 5215-5228. doi: 10.1016/j.na.2012.04.038. [11] V. Calvez and J. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002. [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), 806-841.arXiv:1206.1963. doi: 10.1080/03605302.2014.885046. [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. [14] A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a Chemotaxis Model with Saturated Chemotactic Flux, Kinetic and Related Models, 5 (2012), 51-95. doi: 10.3934/krm.2012.5.51. [15] S. Childress and J. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9. [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), 437-446. doi: 10.1016/j.anihpc.2009.11.016. [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), 5832-5851. doi: 10.1016/j.jde.2012.01.045. [18] M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. [19] M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. [20] E. Feireisl, P. Laurencot and H. Petzeltova, On convergence to equilibria for the Keller-Segel chemotaxis model, J. Differential Equations, 236 (2007), 551-569. doi: 10.1016/j.jde.2007.02.002. [21] H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106. [22] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [23] T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801. doi: 10.1002/mma.569. [24] T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete Coin. Dyn. Syst. Ser. B, 7 (2007), 125-144. doi: 10.3934/dcdsb.2007.7.125. [25] T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [26] D. Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: Some recent results, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 399-423. doi: 10.1007/PL00001455. [27] D. Horstmann, Lyapunov functions and $L^p$-estimates for a class of reaction-diffusion systems, Colloq. Math., 87 (2001), 113-127. doi: 10.4064/cm87-1-7. [28] D. Horstmann, From 1970 until now: The Keller-Segal model in chaemotaxis and its consequence I, Jahresber DMV, 105 (2003), 103-165. [29] D. Horstmann, From 1970 until now: The Keller-Segal model in chaemotaxis and its consequence II, Jahresber DMV, 106 (2004), 51-69. [30] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. [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), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6. [32] J. Jiang and Y. Zhang, On convergence to equilibria for a chemotaxis model with volume-filling effect, Asymptot. Anal., 65 (2009), 79-102. [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), 140-162. doi: 10.1093/imamat/hxl028. [34] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [35] E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. [36] O. Ladyzenskaja, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. [37] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morpholog, J. Theor. Biol., 42 (1973), 63-105 doi: 10.1016/0022-5193(73)90149-5. [38] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. [39] H. Othmer and T. Hillen, The diffusion limit of transport equations. II. Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250. doi: 10.1137/S0036139900382772. [40] B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564. doi: 10.1007/s10492-004-6431-9. [41] C. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. [42] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9. [43] J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009. [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), 790-817. doi: 10.1137/S0036139902415117. [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), 1198-1223. doi: 10.1137/S0036139903433888. [46] X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560. doi: 10.1137/S0036141098339897. [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), 1241-1266. doi: 10.1007/s00285-012-0533-x. [48] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020. [49] T. Xiang, A study on the positive nonconstant steady states of nonlocal chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2457-2485. doi: 10.3934/dcdsb.2013.18.2457. [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), 25-40. doi: 10.1002/mma.1283.

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##### References:
 [1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] N. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113. [3] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. [4] H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence, Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256. [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, No. 44, 32 pp. [6] A. Blanchet, J. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbb{R}^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481. doi: 10.1002/cpa.20225. [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), 133-168. doi: 10.1007/s00526-008-0200-7. [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), 2142-2230. doi: 10.1016/j.jfa.2011.12.012. [9] A. Blanchet, On the Parabolic-Elliptic Patlak-Keller-Segel System in Dimension 2 and Higher, preprint, arXiv:1109.1543. [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), 5215-5228. doi: 10.1016/j.na.2012.04.038. [11] V. Calvez and J. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002. [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), 806-841.arXiv:1206.1963. doi: 10.1080/03605302.2014.885046. [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. [14] A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a Chemotaxis Model with Saturated Chemotactic Flux, Kinetic and Related Models, 5 (2012), 51-95. doi: 10.3934/krm.2012.5.51. [15] S. Childress and J. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9. [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), 437-446. doi: 10.1016/j.anihpc.2009.11.016. [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), 5832-5851. doi: 10.1016/j.jde.2012.01.045. [18] M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. [19] M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. [20] E. Feireisl, P. Laurencot and H. Petzeltova, On convergence to equilibria for the Keller-Segel chemotaxis model, J. Differential Equations, 236 (2007), 551-569. doi: 10.1016/j.jde.2007.02.002. [21] H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106. [22] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [23] T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801. doi: 10.1002/mma.569. [24] T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete Coin. Dyn. Syst. Ser. B, 7 (2007), 125-144. doi: 10.3934/dcdsb.2007.7.125. [25] T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [26] D. Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: Some recent results, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 399-423. doi: 10.1007/PL00001455. [27] D. Horstmann, Lyapunov functions and $L^p$-estimates for a class of reaction-diffusion systems, Colloq. Math., 87 (2001), 113-127. doi: 10.4064/cm87-1-7. [28] D. Horstmann, From 1970 until now: The Keller-Segal model in chaemotaxis and its consequence I, Jahresber DMV, 105 (2003), 103-165. [29] D. Horstmann, From 1970 until now: The Keller-Segal model in chaemotaxis and its consequence II, Jahresber DMV, 106 (2004), 51-69. [30] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. [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), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6. [32] J. Jiang and Y. Zhang, On convergence to equilibria for a chemotaxis model with volume-filling effect, Asymptot. Anal., 65 (2009), 79-102. [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), 140-162. doi: 10.1093/imamat/hxl028. [34] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [35] E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. [36] O. Ladyzenskaja, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. [37] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morpholog, J. Theor. Biol., 42 (1973), 63-105 doi: 10.1016/0022-5193(73)90149-5. [38] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. [39] H. Othmer and T. Hillen, The diffusion limit of transport equations. II. Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250. doi: 10.1137/S0036139900382772. [40] B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564. doi: 10.1007/s10492-004-6431-9. [41] C. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. [42] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9. [43] J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009. [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), 790-817. doi: 10.1137/S0036139902415117. [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), 1198-1223. doi: 10.1137/S0036139903433888. [46] X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560. doi: 10.1137/S0036141098339897. [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), 1241-1266. doi: 10.1007/s00285-012-0533-x. [48] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020. [49] T. Xiang, A study on the positive nonconstant steady states of nonlocal chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2457-2485. doi: 10.3934/dcdsb.2013.18.2457. [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), 25-40. doi: 10.1002/mma.1283.
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