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The existence of strong solutions to the $3D$ Zakharov-Kuznestov equation in a bounded domain
On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model
1. | Department of Mathematics, Tulane University, New Orleans, LA 70118 |
References:
[1] | |
[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. |
show all references
References:
[1] | |
[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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