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Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity
1. | Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130 |
References:
[1] |
A. Ambrosetti and P. Rabinowitz, Dual Variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis, Bioessays, 28 (2006), 9-22.
doi: 10.1002/bies.20343. |
[3] |
P. Biler, Global solutions to some parabolic elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359. |
[4] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Model, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[5] |
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Bioscience, 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[6] |
W.-Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Archive of Rational Mechanics and Analysis, 91 (1986), 283-308.
doi: 10.1007/BF00282336. |
[7] |
D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation, Current Opinion in Genetics Development, 16 (2006), 367-373.
doi: 10.1016/j.gde.2006.06.003. |
[8] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^N$, Advances in Mathematics, Supl Study, 7A (1981), 369-402. |
[9] |
M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Cal. Var. PDE, 11 (2000), 143-175.
doi: 10.1007/PL00009907. |
[10] |
C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Can. J. Math, 52 (2000), 522-538.
doi: 10.4153/CJM-2000-024-x. |
[11] |
M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model, Journal of Mathematical Biology, 35 (1996), 177-194.
doi: 10.1007/s002850050049. |
[12] |
T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[13] |
T. Hillen, K. J. Painter and C. Schmeiser, Global existence for Chemotaxis with finite sampling radius, Discrete Contin. Dyn. Syst-Series B, 7 (2007), 125-144.
doi: 10.3934/dcdsb.2007.7.125. |
[14] |
D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences I, Jahresber DMV, 105 (2003), 103-165. |
[15] |
D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences II, Jahresber DMV, 106 (2004), 51-69. |
[16] |
E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability, Journal of Theoratical Biology, 26 (1970), 399-415. |
[17] |
E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoratical Biology, 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[18] |
E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis, Journal of Theoratical Biology, 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[19] |
M. K. Kwong and L. Zhang, Uniqueness of positive solutions $\Delta u+f(u)=0$ in an annulus, Differential and Intergral Equations, 4 (1991), 583-599. |
[20] |
C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitute stationary solutions to a chemotaxis system, Journal of Differential Equation, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[21] |
R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, Journal of Mathematical Biology, 61 (2010), 739-761.
doi: 10.1007/s00285-009-0317-0. |
[22] |
T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Adv. Math. Soc. Appl, 8 (1998), 145-156. |
[23] |
T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. |
[24] |
T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis, RIMS Kokyuroku, 1009 (1997), 22-28. |
[25] |
V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, Journal. Theor. Biol, 42 (1973), 63-105.
doi: 10.1016/0022-5193(73)90149-5. |
[26] |
W.-M. Ni, Diffusion, cross-diffusion, and their spike layer steady states, Notices of AMS, 45 (1998), 9-18. |
[27] |
W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional. Conf. Ser. Appl. Math. 82. SIAM. Philadelphia, 2011.
doi: 10.1137/1.9781611971972. |
[28] |
W.-M. Ni and I. Takagi, On the shape of least enery solutions to a semilinear Neumann problem, Communication of Pure and Applied Math, 44 (1991), 819-851.
doi: 10.1002/cpa.3160440705. |
[29] |
W.-M. Ni and I. Takagi, Location of the peaks of least energy solutions to a semilinear Neumann problem, Duke Math Journal, 70 (1993), 247-281.
doi: 10.1215/S0012-7094-93-07004-4. |
[30] |
K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial Ekvac, 44 (2001), 441-469. |
[31] |
R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc, 292 (1985), 531-556.
doi: 10.1090/S0002-9947-1985-0808736-1. |
[32] |
B. D. Sleeman, M. J. Ward and J. Wei, The existence, stability, and dynamics of spike patterns in a chemotaxis model, SIAM., Journal of Applied Math, 65 (2005), 790-817.
doi: 10.1137/S0036139902415117. |
[33] |
W. Strauss, Existence of solitary waves in higher dimensions, Communications in Mathematical Physics, 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[34] |
Y. Tao, L. H. Wang and Z. A. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 821-845. |
[35] |
Q. Wang, Global solutions of a Keller-Segel system with saturated logarithmic sensitivity function, Commun. Pure Appl. Anal., 14 (2015), 383-396. |
[36] |
X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM, Journal of Mathematical Analysis, 31 (2000), 535-560.
doi: 10.1137/S0036141098339897. |
[37] |
X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, Journal of Math. Biol, 66 (2013), 1241-1266.
doi: 10.1007/s00285-012-0533-x. |
[38] |
Z. A. Wang, Mathematics of traveling waves in chemotaxis-Review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[39] |
M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[40] |
M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Mathematical Methods in the Applied Sciences, 34 (2011), 176-190.
doi: 10.1002/mma.1346. |
show all references
References:
[1] |
A. Ambrosetti and P. Rabinowitz, Dual Variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis, Bioessays, 28 (2006), 9-22.
doi: 10.1002/bies.20343. |
[3] |
P. Biler, Global solutions to some parabolic elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359. |
[4] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Model, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[5] |
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Bioscience, 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[6] |
W.-Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Archive of Rational Mechanics and Analysis, 91 (1986), 283-308.
doi: 10.1007/BF00282336. |
[7] |
D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation, Current Opinion in Genetics Development, 16 (2006), 367-373.
doi: 10.1016/j.gde.2006.06.003. |
[8] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^N$, Advances in Mathematics, Supl Study, 7A (1981), 369-402. |
[9] |
M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Cal. Var. PDE, 11 (2000), 143-175.
doi: 10.1007/PL00009907. |
[10] |
C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Can. J. Math, 52 (2000), 522-538.
doi: 10.4153/CJM-2000-024-x. |
[11] |
M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model, Journal of Mathematical Biology, 35 (1996), 177-194.
doi: 10.1007/s002850050049. |
[12] |
T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[13] |
T. Hillen, K. J. Painter and C. Schmeiser, Global existence for Chemotaxis with finite sampling radius, Discrete Contin. Dyn. Syst-Series B, 7 (2007), 125-144.
doi: 10.3934/dcdsb.2007.7.125. |
[14] |
D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences I, Jahresber DMV, 105 (2003), 103-165. |
[15] |
D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences II, Jahresber DMV, 106 (2004), 51-69. |
[16] |
E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability, Journal of Theoratical Biology, 26 (1970), 399-415. |
[17] |
E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoratical Biology, 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[18] |
E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis, Journal of Theoratical Biology, 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[19] |
M. K. Kwong and L. Zhang, Uniqueness of positive solutions $\Delta u+f(u)=0$ in an annulus, Differential and Intergral Equations, 4 (1991), 583-599. |
[20] |
C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitute stationary solutions to a chemotaxis system, Journal of Differential Equation, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[21] |
R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, Journal of Mathematical Biology, 61 (2010), 739-761.
doi: 10.1007/s00285-009-0317-0. |
[22] |
T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Adv. Math. Soc. Appl, 8 (1998), 145-156. |
[23] |
T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. |
[24] |
T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis, RIMS Kokyuroku, 1009 (1997), 22-28. |
[25] |
V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, Journal. Theor. Biol, 42 (1973), 63-105.
doi: 10.1016/0022-5193(73)90149-5. |
[26] |
W.-M. Ni, Diffusion, cross-diffusion, and their spike layer steady states, Notices of AMS, 45 (1998), 9-18. |
[27] |
W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional. Conf. Ser. Appl. Math. 82. SIAM. Philadelphia, 2011.
doi: 10.1137/1.9781611971972. |
[28] |
W.-M. Ni and I. Takagi, On the shape of least enery solutions to a semilinear Neumann problem, Communication of Pure and Applied Math, 44 (1991), 819-851.
doi: 10.1002/cpa.3160440705. |
[29] |
W.-M. Ni and I. Takagi, Location of the peaks of least energy solutions to a semilinear Neumann problem, Duke Math Journal, 70 (1993), 247-281.
doi: 10.1215/S0012-7094-93-07004-4. |
[30] |
K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial Ekvac, 44 (2001), 441-469. |
[31] |
R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc, 292 (1985), 531-556.
doi: 10.1090/S0002-9947-1985-0808736-1. |
[32] |
B. D. Sleeman, M. J. Ward and J. Wei, The existence, stability, and dynamics of spike patterns in a chemotaxis model, SIAM., Journal of Applied Math, 65 (2005), 790-817.
doi: 10.1137/S0036139902415117. |
[33] |
W. Strauss, Existence of solitary waves in higher dimensions, Communications in Mathematical Physics, 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[34] |
Y. Tao, L. H. Wang and Z. A. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 821-845. |
[35] |
Q. Wang, Global solutions of a Keller-Segel system with saturated logarithmic sensitivity function, Commun. Pure Appl. Anal., 14 (2015), 383-396. |
[36] |
X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM, Journal of Mathematical Analysis, 31 (2000), 535-560.
doi: 10.1137/S0036141098339897. |
[37] |
X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, Journal of Math. Biol, 66 (2013), 1241-1266.
doi: 10.1007/s00285-012-0533-x. |
[38] |
Z. A. Wang, Mathematics of traveling waves in chemotaxis-Review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[39] |
M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[40] |
M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Mathematical Methods in the Applied Sciences, 34 (2011), 176-190.
doi: 10.1002/mma.1346. |
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