June  2017, 22(4): 1635-1644. doi: 10.3934/dcdsb.2017078

Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author

Received  May 2016 Revised  September 2016 Published  February 2017

Fund Project: Supported by the National Natural Science Foundation of China (11171048) and the Fundamental Research Funds for the Central Universities (DUT16LK24).

This paper deals with the global boundedness of solutions to a fully parabolic Keller-Segel system $u_t=Δ u-\nabla (u^α \nabla v)$, $v_t=Δ v-v+u$ under non-flux boundary conditions in a smooth bounded domain $Ω\subset\mathbb{R}^{n}$. The case of $α≥ \max\{1,\frac{2}{n}\}$ with $n≥1$ was considered in a previous paper of the authors [Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. B, 21 (2016), 1317-1327]. In the present paper we prove for the other case $α∈(\frac{2}{3},1)$ that if $\|u_0\|_{L^\frac{nα}{2}(Ω)}$ and $\|\nabla v_0\|_{L^{nα}(Ω)}$ are small enough with $n≥q3$, then the solutions are globally bounded with both $u$ and $v$ decaying to the same constant steady state $\bar{u}_0=\frac{1}{|Ω|}∈t_Ω u_0(x) dx$ exponentially in the $L^∞$-norm as $t? ∞$. Moreover, the above conclusions still hold for all $α≥q2$ and $n≥q1$, provided $\|u_0\|_{L^{nα-n}(Ω)}$ and $\|\nabla v_0\|_{L^{∞}(Ω)}$ sufficiently small.

Citation: Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078
References:
[1]

N. BellomoA. BellouquidY. Tao and M. Winker, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

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X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar

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T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.  Google Scholar

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T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., 129 (2014), 135-146.  doi: 10.1007/s10440-013-9832-5.  Google Scholar

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T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.  Google Scholar

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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.  Google Scholar

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S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel system of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

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S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate keller-Segel system of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596.  doi: 10.3934/dcdsb.2013.18.2569.  Google Scholar

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

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N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaé Anal. Non Linéaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.  Google Scholar

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A. Montaru, A semilinear parabolic-elliptic chemotaxis system with critical mass in any space dimension, Nonlinearity, 26 (2013), 2669-2701.  doi: 10.1088/0951-7715/26/9/2669.  Google Scholar

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A. Montaru, Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 231-256.  doi: 10.3934/dcdsb.2014.19.231.  Google Scholar

[13]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[14]

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.  Google Scholar

[15]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.  Google Scholar

[16]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.  Google Scholar

[17]

H. YuW. Wang and S. Zheng, Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. B, 21 (2016), 1317-1327.  doi: 10.3934/dcdsb.2016.21.1317.  Google Scholar

show all references

References:
[1]

N. BellomoA. BellouquidY. Tao and M. Winker, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

[2]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar

[3]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.  Google Scholar

[4]

T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., 129 (2014), 135-146.  doi: 10.1007/s10440-013-9832-5.  Google Scholar

[5]

T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.  Google Scholar

[6]

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.  Google Scholar

[7]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel system of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

[8]

S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate keller-Segel system of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596.  doi: 10.3934/dcdsb.2013.18.2569.  Google Scholar

[9]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[10]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaé Anal. Non Linéaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.  Google Scholar

[11]

A. Montaru, A semilinear parabolic-elliptic chemotaxis system with critical mass in any space dimension, Nonlinearity, 26 (2013), 2669-2701.  doi: 10.1088/0951-7715/26/9/2669.  Google Scholar

[12]

A. Montaru, Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 231-256.  doi: 10.3934/dcdsb.2014.19.231.  Google Scholar

[13]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[14]

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.  Google Scholar

[15]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.  Google Scholar

[16]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.  Google Scholar

[17]

H. YuW. Wang and S. Zheng, Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. B, 21 (2016), 1317-1327.  doi: 10.3934/dcdsb.2016.21.1317.  Google Scholar

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