Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces

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May 2015, 35(5): 1891-1904. doi: 10.3934/dcds.2015.35.1891

Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces

1.	School of Mathematical Science, Dalian University of Technology, Dalian 116023, China

Received May 2014 Revised September 2014 Published December 2014

Abstract
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In this paper, the fully parabolic Keller-Segel system \begin{equation}\label{problemAbstract}\left\{\begin{array}{ll} u_t=\Delta u-\nabla\cdot(u\nabla v), &(x,t)\in \Omega\times (0,T),\\ v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T),\\ \end{array}\right.\tag{$\star$} \end{equation} is considered under Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^n$ with smooth boundary, where $n\ge 2$. We derive a smallness condition on the initial data in optimal Lebesgue spaces which ensure global boundedness and large time convergence. More precisely, we shall show that one can find $\varepsilon_0>0$ such that for all suitably regular initial data $(u_0,v_0)$ satisfying $\|u_0\|_{L^{\frac{n}{2}}(\Omega)}<\varepsilon_0$ and $\|\nabla v_0\|_{L^{n}(\Omega)}<\varepsilon_0$, the above problem possesses a global classical solution which is bounded and converges to the constant steady state $(m,m)$ with $m:=\frac{1}{|\Omega|}\int_\Omega u_0$.
Our approach allows us to furthermore study a general chemotaxis system with rotational sensitivity in dimension 2, which is lacking the natural energy structure associated with ($\star$). For such systems, we prove a global existence and boundedness result under corresponding smallness conditions on the initially present total mass of cells and the chemical gradient.

Keywords: boundedness, Chemotaxis, Keller-Segel system, long time behavior., fixed point theory.

Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 92C1.

Citation: Xinru Cao. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1891-1904. doi: 10.3934/dcds.2015.35.1891

References:

[1]	X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation, Nonlinearity, 27 (2014), 1899-1913. doi: 10.1088/0951-7715/27/8/1899.
[2]	L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math, 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x.
[3]	L. Corrias and B. Perthame, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces, Mathematical and Computer Modelling, 47 (2008), 755-764. doi: 10.1016/j.mcm.2007.06.005.
[4]	M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.
[5]	T. Hillen and K. J. Painter, A user's guide to PDE models in a chemotaxis, J. Math. Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.
[6]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jber. DMV, 105 (2003), 103-165.
[7]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.
[8]	W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Tran. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.
[9]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[10]	O. A. Ladyžxenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R. I.,1968.
[11]	T. Li, A. Suen, C. Xue and M. Winkler, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., DOI: 10.1142/S0218202515500177.
[12]	T. Nagai, Blowup of Nonradial Solutions to Parabolic-Elliptic Systems Modeling Chemotaxis in Two-Dimensional Domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.
[13]	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.
[14]	R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl. 2, North-Hlland, Amsterdam, 1997. doi: 10.1115/1.3424338.
[15]	I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102.
[16]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.
[17]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[18]	C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population, SIAM J. Appl. Math., 70 (2009), 133-167. doi: 10.1137/070711505.

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References:

[1]	X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation, Nonlinearity, 27 (2014), 1899-1913. doi: 10.1088/0951-7715/27/8/1899.
[2]	L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math, 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x.
[3]	L. Corrias and B. Perthame, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces, Mathematical and Computer Modelling, 47 (2008), 755-764. doi: 10.1016/j.mcm.2007.06.005.
[4]	M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.
[5]	T. Hillen and K. J. Painter, A user's guide to PDE models in a chemotaxis, J. Math. Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.
[6]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jber. DMV, 105 (2003), 103-165.
[7]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.
[8]	W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Tran. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.
[9]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[10]	O. A. Ladyžxenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R. I.,1968.
[11]	T. Li, A. Suen, C. Xue and M. Winkler, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., DOI: 10.1142/S0218202515500177.
[12]	T. Nagai, Blowup of Nonradial Solutions to Parabolic-Elliptic Systems Modeling Chemotaxis in Two-Dimensional Domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.
[13]	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.
[14]	R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl. 2, North-Hlland, Amsterdam, 1997. doi: 10.1115/1.3424338.
[15]	I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102.
[16]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.
[17]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[18]	C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population, SIAM J. Appl. Math., 70 (2009), 133-167. doi: 10.1137/070711505.

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