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

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.
Citation: Xinru Cao. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1088/0951-7715/27/8/1899. Google Scholar

[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. doi: 10.1007/s00032-003-0026-x. Google Scholar

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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. doi: 10.1016/j.mcm.2007.06.005. Google Scholar

[4]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Normale Superiore, 24 (1997), 633. Google Scholar

[5]

T. Hillen and K. J. Painter, A user's guide to PDE models in a chemotaxis,, J. Math. Biology, 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

[6]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, Jber. DMV, 105 (2003), 103. Google Scholar

[7]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022. Google Scholar

[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. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar

[9]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[10]

O. A. Ladyžxenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society, (). Google Scholar

[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., (). Google Scholar

[12]

T. Nagai, Blowup of Nonradial Solutions to Parabolic-Elliptic Systems Modeling Chemotaxis in Two-Dimensional Domains,, J. Inequal. Appl., 6 (2001), 37. doi: 10.1155/S1025583401000042. Google Scholar

[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. Google Scholar

[14]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Stud. Math. Appl. 2, 2 (1997). doi: 10.1115/1.3424338. Google Scholar

[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. doi: 10.1073/pnas.0406724102. Google Scholar

[16]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar

[17]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748. doi: 10.1016/j.matpur.2013.01.020. Google Scholar

[18]

C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population,, SIAM J. Appl. Math., 70 (2009), 133. doi: 10.1137/070711505. Google Scholar

show all references

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. doi: 10.1088/0951-7715/27/8/1899. Google Scholar

[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. doi: 10.1007/s00032-003-0026-x. Google Scholar

[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. doi: 10.1016/j.mcm.2007.06.005. Google Scholar

[4]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Normale Superiore, 24 (1997), 633. Google Scholar

[5]

T. Hillen and K. J. Painter, A user's guide to PDE models in a chemotaxis,, J. Math. Biology, 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

[6]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, Jber. DMV, 105 (2003), 103. Google Scholar

[7]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022. Google Scholar

[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. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar

[9]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[10]

O. A. Ladyžxenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society, (). Google Scholar

[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., (). Google Scholar

[12]

T. Nagai, Blowup of Nonradial Solutions to Parabolic-Elliptic Systems Modeling Chemotaxis in Two-Dimensional Domains,, J. Inequal. Appl., 6 (2001), 37. doi: 10.1155/S1025583401000042. Google Scholar

[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. Google Scholar

[14]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Stud. Math. Appl. 2, 2 (1997). doi: 10.1115/1.3424338. Google Scholar

[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. doi: 10.1073/pnas.0406724102. Google Scholar

[16]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar

[17]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748. doi: 10.1016/j.matpur.2013.01.020. Google Scholar

[18]

C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population,, SIAM J. Appl. Math., 70 (2009), 133. doi: 10.1137/070711505. Google Scholar

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