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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] |
R. Adams, Sobolev Spaces,, Academic Press, (1975).
|
[2] |
N. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827.
doi: 10.1080/03605307908820113. |
[3] |
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13.
|
[4] |
H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence,, Math. Z., 202 (1989), 219.
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, (2006).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
S. Childress and J. Percus, Nonlinear aspects of chemotaxis,, Math. Biosci., 56 (1981), 217.
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.
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.
doi: 10.1016/j.jde.2012.01.045. |
[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.
|
[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. |
[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. |
[22] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[28] |
D. Horstmann, From 1970 until now: The Keller-Segal model in chaemotaxis and its consequence I,, Jahresber DMV, 105 (2003), 103.
|
[29] |
D. Horstmann, From 1970 until now: The Keller-Segal model in chaemotaxis and its consequence II,, Jahresber DMV, 106 (2004), 51.
|
[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. |
[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. |
[32] |
J. Jiang and Y. Zhang, On convergence to equilibria for a chemotaxis model with volume-filling effect,, Asymptot. Anal., 65 (2009), 79.
|
[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. |
[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. |
[35] |
E. Keller and L. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225.
doi: 10.1016/0022-5193(71)90050-6. |
[36] |
O. Ladyzenskaja, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type,, AMS, (1968).
|
[37] |
V. Nanjundiah, Chemotaxis, signal relaying and aggregation morpholog,, J. Theor. Biol., 42 (1973), 63.
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.
|
[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. |
[40] |
B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539.
doi: 10.1007/s10492-004-6431-9. |
[41] |
C. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311.
doi: 10.1007/BF02476407. |
[42] |
P. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Funct. Anal., 7 (1971), 487.
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.
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.
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.
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.
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.
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.
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.
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.
doi: 10.1002/mma.1283. |
show all references
References:
[1] |
R. Adams, Sobolev Spaces,, Academic Press, (1975).
|
[2] |
N. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827.
doi: 10.1080/03605307908820113. |
[3] |
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13.
|
[4] |
H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence,, Math. Z., 202 (1989), 219.
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, (2006).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
S. Childress and J. Percus, Nonlinear aspects of chemotaxis,, Math. Biosci., 56 (1981), 217.
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.
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.
doi: 10.1016/j.jde.2012.01.045. |
[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.
|
[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. |
[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. |
[22] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[28] |
D. Horstmann, From 1970 until now: The Keller-Segal model in chaemotaxis and its consequence I,, Jahresber DMV, 105 (2003), 103.
|
[29] |
D. Horstmann, From 1970 until now: The Keller-Segal model in chaemotaxis and its consequence II,, Jahresber DMV, 106 (2004), 51.
|
[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. |
[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. |
[32] |
J. Jiang and Y. Zhang, On convergence to equilibria for a chemotaxis model with volume-filling effect,, Asymptot. Anal., 65 (2009), 79.
|
[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. |
[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. |
[35] |
E. Keller and L. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225.
doi: 10.1016/0022-5193(71)90050-6. |
[36] |
O. Ladyzenskaja, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type,, AMS, (1968).
|
[37] |
V. Nanjundiah, Chemotaxis, signal relaying and aggregation morpholog,, J. Theor. Biol., 42 (1973), 63.
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.
|
[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. |
[40] |
B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539.
doi: 10.1007/s10492-004-6431-9. |
[41] |
C. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311.
doi: 10.1007/BF02476407. |
[42] |
P. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Funct. Anal., 7 (1971), 487.
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.
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.
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.
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.
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.
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.
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.
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.
doi: 10.1002/mma.1283. |
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