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Numerical simulation of chemotaxis models on stationary surfaces
Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity
1. | Department of Applied Mathematics, Dong Hua University, Shanghai 200051 |
It is proved that under the assumptions that $0\not\equiv u_0\in C^0(\bar{\Omega})$ and $v_0\in C^1(\bar{\Omega})$ are nonnegative and that $\alpha<\frac{4}{n+2}$, the classical solutions to the above system are uniformly-in-time bounded. Moreover, it is shown that for any given $(u_0, v_0)$, the corresponding solution $(u, v)$ converges to $(\bar{u}_0, \bar{u}_0)$ as time goes to infinity, where $\bar{u}_0 :=\frac{1}{\Omega} f_{\Omega} u_0$.
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Nonlinearity, 24 (2011), 1253-1270.
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show all references
References:
[1] |
Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
Lect. 2nd 1985 Sess. C. I. M. E., Montecatini Terme/Italy 1985, Lect. Notes Math., 1224 (1986), 1-46.
doi: 10.1007/BFb0072687. |
[3] |
Dev. Biol., 296 (2006), 137-149.
doi: 10.1016/j.ydbio.2006.04.451. |
[4] |
Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 437-446.
doi: 10.1016/j.anihpc.2009.11.016. |
[5] |
Banach Center Publ., 81 (2008), Polish Acad. Sci., Warsaw, 105-117.
doi: 10.4064/bc81-0-7. |
[6] |
Holt, Rinehart & Winston, New York, 1969. |
[7] |
Euro. J. Neuroscicen, 19 (2004), 831-844. Google Scholar |
[8] |
Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977. |
[9] |
Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. |
[10] |
Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683. |
[11] |
J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[12] |
Jahresber. Deutsch. Math.- Verien, 105 (2003), 103-165. |
[13] |
European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[14] |
J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[15] |
Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.2307/2153966. |
[16] |
J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[17] |
J. Math. Anal. Appl., 343 (2008), 379-398.
doi: 10.1016/j.jmaa.2008.01.005. |
[18] |
Arch. Rational Mech. Anal., 74 (1980), 335-353.
doi: 10.1007/BF00249679. |
[19] |
Bull. Math. Biol., 65 (2003), 673-730. Google Scholar |
[20] |
J. of Inequal. & Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[21] |
Can. Appl. Math. Quart., 10 (2002), 501-543. |
[22] |
Nonlinearity, 24 (2011), 1253-1270.
doi: 10.1088/0951-7715/24/4/012. |
[23] |
Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[24] |
J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[25] |
J. Differential Equations, 252 (2012), 2520-2534.
doi: 10.1016/j.jde.2011.07.010. |
[26] |
Ann. Inst. H. Poincaré, Analyse Non Linéaire, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[27] |
Math. Methods Appl. Sci., 25 (2002), 911-925.
doi: 10.1002/mma.319. |
[28] |
J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[29] |
J. Math. Pures Appl., 99 (2013).
doi: 10.1016/j.matpur.2013.01.020. |
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