January  2016, 21(1): 81-102. doi: 10.3934/dcdsb.2016.21.81

Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity

1. 

Department of Mathematics, Tokyo University of Science, Tokyo 162-8601

2. 

Department of Mathematics, Kyushu Institute of Technology, Sensuicho, Tobata, Kitakyushu 804-8550

Received  June 2015 Revised  July 2015 Published  November 2015

This paper is concerned with the parabolic-elliptic Keller-Segel system with signal-dependent sensitivity $\chi(v)$, \begin{align*} \begin{cases} u_t=\Delta u - \nabla \cdot ( u \nabla \chi(v)) &\mathrm{in}\ \Omega\times(0,\infty), \\ 0=\Delta v -v+u &\mathrm{in}\ \Omega\times(0,\infty), \end{cases} \end{align*} under homogeneous Neumann boundary condition in a smoothly bounded domain $\Omega \subset \mathbb{R}^2$ with nonnegative initial data $u_0 \in C^{0}(\overline{\Omega})$, $\not\equiv 0$.
    In the special case $\chi(v)=\chi_0 \log v\, (\chi_0>0)$, global existence and boundedness of the solution to the system were proved under some smallness condition on $\chi_0$ by Biler (1999) and Fujie, Winkler and Yokota (2015). In the present work, global existence and boundedness in the system will be established for general sensitivity $\chi$ satisfying $\chi'>0$ and $\chi'(s) \to 0 $ as $s\to \infty$. In particular, this establishes global existence and boundedness in the case $\chi(v)=\chi_0\log v$ with large $\chi_0>0$. Moreover, although the methods in the previous results are effective for only few specific cases, the present method can be applied to more general cases requiring only the essential conditions. Actually, our condition is necessary, since there are many radial blow-up solutions in the case $\inf_{s>0} \chi^\prime (s) >0$.
Citation: Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81
References:
[1]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis,, Adv. Math. Sci. Appl., 9 (1999), 347.   Google Scholar

[2]

H. Brézis and W. Strauss, Semi-linear second-order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565.  doi: 10.2969/jmsj/02540565.  Google Scholar

[3]

S. Y. A. Chang and P. Yang, Conformal deformation of metrics on $S^2$,, J. Differential Geom., 27 (1988), 259.   Google Scholar

[4]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity,, J. Math. Anal. Appl., 424 (2015), 675.  doi: 10.1016/j.jmaa.2014.11.045.  Google Scholar

[5]

K. Fujie, M. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity,, Math. Methods Appl. Sci., 38 (2015), 1212.  doi: 10.1002/mma.3149.  Google Scholar

[6]

K. Fujie and T. Yokota, Boundedness in a fully parabolic chemotaxis system with strongly singular sensitivity,, Appl. Math. Lett., 38 (2014), 140.  doi: 10.1016/j.aml.2014.07.021.  Google Scholar

[7]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 663.   Google Scholar

[8]

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

[9]

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

[10]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819.  doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[11]

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

[12]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis,, J. Theor. Biol., 30 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[13]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.   Google Scholar

[14]

T. Nagai, Blow-up 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

[15]

T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 145.   Google Scholar

[16]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkc. Ekvacioj, 40 (1997), 411.   Google Scholar

[17]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, J. Theor. Biol., 42 (1973), 63.  doi: 10.1016/0022-5193(73)90149-5.  Google Scholar

[18]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology,, Adv. Differential Equations, 6 (2001), 21.   Google Scholar

[19]

Y. Sugiyama, On $\varepsilon$-regularity theorem and asymptotic behaviors of solutions for Keller-Segel systems,, SIAM J. Math. Anal., 41 (2009), 1664.  doi: 10.1137/080721078.  Google Scholar

[20]

M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity,, Math. Nachr., 283 (2010), 1664.  doi: 10.1002/mana.200810838.  Google Scholar

[21]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Math. Methods Appl. Sci., 34 (2011), 176.  doi: 10.1002/mma.1346.  Google Scholar

show all references

References:
[1]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis,, Adv. Math. Sci. Appl., 9 (1999), 347.   Google Scholar

[2]

H. Brézis and W. Strauss, Semi-linear second-order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565.  doi: 10.2969/jmsj/02540565.  Google Scholar

[3]

S. Y. A. Chang and P. Yang, Conformal deformation of metrics on $S^2$,, J. Differential Geom., 27 (1988), 259.   Google Scholar

[4]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity,, J. Math. Anal. Appl., 424 (2015), 675.  doi: 10.1016/j.jmaa.2014.11.045.  Google Scholar

[5]

K. Fujie, M. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity,, Math. Methods Appl. Sci., 38 (2015), 1212.  doi: 10.1002/mma.3149.  Google Scholar

[6]

K. Fujie and T. Yokota, Boundedness in a fully parabolic chemotaxis system with strongly singular sensitivity,, Appl. Math. Lett., 38 (2014), 140.  doi: 10.1016/j.aml.2014.07.021.  Google Scholar

[7]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 663.   Google Scholar

[8]

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

[9]

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

[10]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819.  doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[11]

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

[12]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis,, J. Theor. Biol., 30 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[13]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.   Google Scholar

[14]

T. Nagai, Blow-up 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

[15]

T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 145.   Google Scholar

[16]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkc. Ekvacioj, 40 (1997), 411.   Google Scholar

[17]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, J. Theor. Biol., 42 (1973), 63.  doi: 10.1016/0022-5193(73)90149-5.  Google Scholar

[18]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology,, Adv. Differential Equations, 6 (2001), 21.   Google Scholar

[19]

Y. Sugiyama, On $\varepsilon$-regularity theorem and asymptotic behaviors of solutions for Keller-Segel systems,, SIAM J. Math. Anal., 41 (2009), 1664.  doi: 10.1137/080721078.  Google Scholar

[20]

M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity,, Math. Nachr., 283 (2010), 1664.  doi: 10.1002/mana.200810838.  Google Scholar

[21]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Math. Methods Appl. Sci., 34 (2011), 176.  doi: 10.1002/mma.1346.  Google Scholar

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