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Trace Hardy--Sobolev--Maz'ya inequalities for the half fractional Laplacian
Global solutions of a Keller--Segel system with saturated logarithmic sensitivity function
1. | Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130 |
References:
[1] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, Teubner, Stuttgart, Leipzig, (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[2] |
P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Advances in Mathematical Sciences and Applications, 9 (1999), 347-359. |
[3] |
M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis, Bioessays, 28 (2006), 9-22. |
[4] |
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Bioscience, 56 (1983), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[5] |
D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation, Current Opinion in Genetics Development, 16 (2006), 367-373. |
[6] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag-Berlin-New York, 1981. |
[7] |
D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences I, Jahresber DMV, 105 (2003), 103-165. |
[8] |
D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences II, Jahresber DMV, 106 (2003), 51-69. |
[9] |
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. |
[10] |
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. |
[11] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxisi system, J. Diff. Equation, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[12] |
M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model, Journal of Mathematical Biology, 35 (1996), 583-623.
doi: 10.1007/s002850050049. |
[13] |
E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability, Journal of Theoratical Biology, 26 (1970), 399-415. |
[14] |
E. F. Keller and L. A. Segel, Model for Chemotaxis, Journal of Theoratical Biology, 30 (1971), 225-234. |
[15] |
E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis, Journal of Theoratical Biology, 30 (1971), 235-248. |
[16] |
T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541.
doi: 10.1137/09075161X. |
[17] |
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. |
[18] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968. |
[19] |
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 (1997), 145-156. |
[20] |
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. |
[21] |
T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis, RIMS Kokyuroku, 1009 (1997), 22-28. |
[22] |
V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, Journal. Theor. Biol., 42 (1973), 63-105. |
[23] |
K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial Ekvac, 44 (2001), 441-469. |
[24] |
C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Analysis: Real World Applications, 12 (2011), 3727-3740.
doi: 10.1016/j.nonrwa.2011.07.006. |
[25] |
Z. A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst.-Series B, 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[26] |
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. |
[27] |
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. |
[28] |
M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr, 283 (2010), 1664-1673.
doi: 10.1002/mana.200810838. |
[29] |
A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Jap, 45 (1997), 241-265. |
show all references
References:
[1] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, Teubner, Stuttgart, Leipzig, (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[2] |
P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Advances in Mathematical Sciences and Applications, 9 (1999), 347-359. |
[3] |
M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis, Bioessays, 28 (2006), 9-22. |
[4] |
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Bioscience, 56 (1983), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[5] |
D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation, Current Opinion in Genetics Development, 16 (2006), 367-373. |
[6] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag-Berlin-New York, 1981. |
[7] |
D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences I, Jahresber DMV, 105 (2003), 103-165. |
[8] |
D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences II, Jahresber DMV, 106 (2003), 51-69. |
[9] |
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. |
[10] |
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. |
[11] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxisi system, J. Diff. Equation, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[12] |
M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model, Journal of Mathematical Biology, 35 (1996), 583-623.
doi: 10.1007/s002850050049. |
[13] |
E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability, Journal of Theoratical Biology, 26 (1970), 399-415. |
[14] |
E. F. Keller and L. A. Segel, Model for Chemotaxis, Journal of Theoratical Biology, 30 (1971), 225-234. |
[15] |
E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis, Journal of Theoratical Biology, 30 (1971), 235-248. |
[16] |
T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541.
doi: 10.1137/09075161X. |
[17] |
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. |
[18] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968. |
[19] |
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 (1997), 145-156. |
[20] |
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. |
[21] |
T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis, RIMS Kokyuroku, 1009 (1997), 22-28. |
[22] |
V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, Journal. Theor. Biol., 42 (1973), 63-105. |
[23] |
K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial Ekvac, 44 (2001), 441-469. |
[24] |
C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Analysis: Real World Applications, 12 (2011), 3727-3740.
doi: 10.1016/j.nonrwa.2011.07.006. |
[25] |
Z. A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst.-Series B, 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[26] |
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. |
[27] |
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. |
[28] |
M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr, 283 (2010), 1664-1673.
doi: 10.1002/mana.200810838. |
[29] |
A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Jap, 45 (1997), 241-265. |
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