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A study on the positive nonconstant steady states of nonlocal chemotaxis systems
1. | Department of Mathematics, Tulane University, New Orleans, LA 70118, United States |
References:
[1] |
M. Burger, Y. Dolak-Struss and C. Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions, Communications in Mathematical Sciences, 6 (2008), 1-28.
doi: 10.4310/CMS.2008.v6.n1.a1. |
[2] |
X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, On Keller-Segel's minimal chemotaxis model,, in process., ().
|
[3] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models (KRM), 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[4] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[5] |
Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM Journal on Applied Mathematics, 66 (2005), 286-308.
doi: 10.1137/040612841. |
[6] |
A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subject to chemotaxis, Mathematical Models and Methods in Applied Sciences, 14 (2004), 503-533.
doi: 10.1142/S0218202504003337. |
[7] |
P. Grindrod, J. Murray and S. Sinha, Steady-state spatial patterns in a cell-chemotaxis model, IMA J. Math. Appl. Med. Biol., 6 (1989), 69-79.
doi: 10.1093/imammb/6.2.69. |
[8] |
T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete and Continuous Dynamical Systems - Series B (DCDS-B), 7 (2007), 125-144.
doi: 10.3934/dcdsb.2007.7.125. |
[9] |
T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[10] |
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber DMV, 105 (2003), 103-165. |
[11] |
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences II, Jahresber DMV, 106 (2004), 51-69. |
[12] |
E. Keller and L. 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. |
[13] |
E. Keller and L. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[14] |
C. Lin, W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, Journal of Differential Equations, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[15] |
M. Myerscough, P. Maini and K. Painter, Pattern formation in a generalized chemo-tactic model, Bull. Math. Biol., 60 (1998), 1-26. |
[16] |
W. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Mathematical Journal, 70 (1993), 247-281.
doi: 10.1215/S0012-7094-93-07004-4. |
[17] |
W. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. |
[18] |
H. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM Journal on Applied Mathematics, 62 (2002), 1222-1250.
doi: 10.1137/S0036139900382772. |
[19] |
K. Painter, H. Othmer and P. Maini, Stripe formation in juvenile pomacanthus via chemotactic response to a reaction-diffusion mechanism, Proc. Natl. Acad. Sci. USA, 96 (1999), 5549-5554. |
[20] |
C. Patlak, Random walk with persistence and external bias, Bulletin Mathematical Biophysics, 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[21] |
J. Pejsachowicz and P. Rabier, Degree theory for C1 Fredholm mappings of index 0, Journal d'Analyse Mathematique, 76 (1998), 289-319.
doi: 10.1007/BF02786939. |
[22] |
A. Potapov, T. Hillen, Metastability in chemotaxis models, Journal of Dynamics and Differential Equations, 17 (2005), 293-330.
doi: 10.1007/s10884-005-2938-3. |
[23] |
P. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202.
doi: 10.1216/RMJ-1973-3-2-161. |
[24] |
P. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[25] |
R. Schaaf, Stationary solutions of chemotaxis systems, Transactions of the American Mathematical Society, 292 (1985), 531-556.
doi: 10.1090/S0002-9947-1985-0808736-1. |
[26] |
J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, Journal of Differential Equations, 246 (2009), 2788-2812.
doi: 10.1016/j.jde.2008.09.009. |
[27] |
R. Tyson, S. R. Lubkin and J. D. Murray, A minimal mechanism for bacterial pattern formation, Proceedings of the Royal Society, 266 (1999), 299-304.
doi: 10.1098/rspb.1999.0637. |
[28] |
X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM Journal on Mathematical Analysis, 31 (2000), 535-560.
doi: 10.1137/S0036141098339897. |
[29] |
X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation method and Helly's compactness theorem, Journal of Mathematical Biology, 66 (2013), 1241-1266.
doi: 10.1007/s00285-012-0533-x. |
[30] |
J. Wei, Existence and stability of spikes for the Gierer-Meinhardt system, Handbook of Differential Equations: Stationary Partial Differential Equations, 5 (2008), 487-585.
doi: 10.1016/S1874-5733(08)80013-7. |
show all references
References:
[1] |
M. Burger, Y. Dolak-Struss and C. Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions, Communications in Mathematical Sciences, 6 (2008), 1-28.
doi: 10.4310/CMS.2008.v6.n1.a1. |
[2] |
X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, On Keller-Segel's minimal chemotaxis model,, in process., ().
|
[3] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models (KRM), 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[4] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[5] |
Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM Journal on Applied Mathematics, 66 (2005), 286-308.
doi: 10.1137/040612841. |
[6] |
A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subject to chemotaxis, Mathematical Models and Methods in Applied Sciences, 14 (2004), 503-533.
doi: 10.1142/S0218202504003337. |
[7] |
P. Grindrod, J. Murray and S. Sinha, Steady-state spatial patterns in a cell-chemotaxis model, IMA J. Math. Appl. Med. Biol., 6 (1989), 69-79.
doi: 10.1093/imammb/6.2.69. |
[8] |
T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete and Continuous Dynamical Systems - Series B (DCDS-B), 7 (2007), 125-144.
doi: 10.3934/dcdsb.2007.7.125. |
[9] |
T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[10] |
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber DMV, 105 (2003), 103-165. |
[11] |
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences II, Jahresber DMV, 106 (2004), 51-69. |
[12] |
E. Keller and L. 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. |
[13] |
E. Keller and L. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[14] |
C. Lin, W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, Journal of Differential Equations, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[15] |
M. Myerscough, P. Maini and K. Painter, Pattern formation in a generalized chemo-tactic model, Bull. Math. Biol., 60 (1998), 1-26. |
[16] |
W. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Mathematical Journal, 70 (1993), 247-281.
doi: 10.1215/S0012-7094-93-07004-4. |
[17] |
W. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. |
[18] |
H. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM Journal on Applied Mathematics, 62 (2002), 1222-1250.
doi: 10.1137/S0036139900382772. |
[19] |
K. Painter, H. Othmer and P. Maini, Stripe formation in juvenile pomacanthus via chemotactic response to a reaction-diffusion mechanism, Proc. Natl. Acad. Sci. USA, 96 (1999), 5549-5554. |
[20] |
C. Patlak, Random walk with persistence and external bias, Bulletin Mathematical Biophysics, 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[21] |
J. Pejsachowicz and P. Rabier, Degree theory for C1 Fredholm mappings of index 0, Journal d'Analyse Mathematique, 76 (1998), 289-319.
doi: 10.1007/BF02786939. |
[22] |
A. Potapov, T. Hillen, Metastability in chemotaxis models, Journal of Dynamics and Differential Equations, 17 (2005), 293-330.
doi: 10.1007/s10884-005-2938-3. |
[23] |
P. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202.
doi: 10.1216/RMJ-1973-3-2-161. |
[24] |
P. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[25] |
R. Schaaf, Stationary solutions of chemotaxis systems, Transactions of the American Mathematical Society, 292 (1985), 531-556.
doi: 10.1090/S0002-9947-1985-0808736-1. |
[26] |
J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, Journal of Differential Equations, 246 (2009), 2788-2812.
doi: 10.1016/j.jde.2008.09.009. |
[27] |
R. Tyson, S. R. Lubkin and J. D. Murray, A minimal mechanism for bacterial pattern formation, Proceedings of the Royal Society, 266 (1999), 299-304.
doi: 10.1098/rspb.1999.0637. |
[28] |
X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM Journal on Mathematical Analysis, 31 (2000), 535-560.
doi: 10.1137/S0036141098339897. |
[29] |
X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation method and Helly's compactness theorem, Journal of Mathematical Biology, 66 (2013), 1241-1266.
doi: 10.1007/s00285-012-0533-x. |
[30] |
J. Wei, Existence and stability of spikes for the Gierer-Meinhardt system, Handbook of Differential Equations: Stationary Partial Differential Equations, 5 (2008), 487-585.
doi: 10.1016/S1874-5733(08)80013-7. |
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