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October  2019, 24(10): 5297-5315. doi: 10.3934/dcdsb.2019059

$ L^\sigma $-measure criteria for boundedness in a quasilinear parabolic-parabolic Keller-Segel system with supercritical sensitivity

Mengyao Ding 1,, and  Xiangdong Zhao 2,

1. 

School of Mathematical Sciences, Peking University, Beijing, 100871, China
2. 

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

* Corresponding author: Mengyao Ding

Received  March 2018 Revised  October 2018 Published  October 2019 Early access  April 2019

Fund Project: The first author is supported by the National Natural Science Foundation of China (11571020, 11671021).

This paper studies the parabolic-parabolic Keller-Segel system with supercritical sensitivity: $u_{t}=\nabla\cdot(\phi (u) \nabla u)-\nabla \cdot(\varphi(u)\nabla v)$, $v_{t}=\Delta v -v+u$, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain $\Omega\subset\mathbb{R}^n$ $(n\ge2)$, the diffusivity fulfills $\phi(u)\ge a_0(u+1)^{\gamma}$ with $\gamma\ge0$ and $a_0>0$, while the chemotactic sensitivity satisfies $0\le \varphi(u)\le b_0u(u+1)^{\alpha+\gamma-1}$ with $\alpha>\frac{2}{n}$ and $b_0>0$. It is proved that the problem possesses a globally bounded solution for $\frac{4}{n+2}<\alpha<2$, whenever $\|u_0\|_{L^{\frac{n\alpha}{2}}(\Omega)}$ and $\|\nabla v_0\|_{L^{\frac{n\alpha+2\gamma}{2-\alpha}}(\Omega)}$ is sufficiently small. Similarly, the above conclusion still holds for $\alpha>2$ provided that $\|u_{0}\|_{L^{n\alpha-n}(\Omega)}$ and $\|\nabla v_0\|_{L^{\infty}(\Omega)}$ are small enough.

Keywords: Keller-Segel system, nonlinear diffusion, supercritical sensitivity, small initial data, boundedness.
Mathematics Subject Classification: Primary: 35K55, 35B35; Secondary: 92C17.
Citation: Mengyao Ding, Xiangdong Zhao. $ L^\sigma $-measure criteria for boundedness in a quasilinear parabolic-parabolic Keller-Segel system with supercritical sensitivity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5297-5315. doi: 10.3934/dcdsb.2019059
References:
[1]

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.

[2]

T. Cieślak and C. Morales-Rodrigo, Quasilinear non-uniformly parabolic-elliptic system modelling chemotaxis with volume filling effect: Existence and uniqueness of global-in-time solutions, Topol. Methods Nonlinear Anal., 29 (2007), 361-381. 

[3]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.

[4]

T. Cieślak and M. Winkler, Global bounded solutions in a two-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal. Real World Appl., 35 (2017), 1-19.  doi: 10.1016/j.nonrwa.2016.10.002.

[5]

T. Cieślak and M. Winkler, Stabilization in a higher-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal., 159 (2017), 129-144.  doi: 10.1016/j.na.2016.04.013.

[6]

M. Ding and S. Zheng, $ L^\gamma $-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity, Discrete Contin. Dyn. Syst. Ser. B, Online First, (2018). doi: 10.3934/dcdsb.2018295.

[7]

L. Fan and H. Jin, Global existence and asymptotic behavior to a chemotaxis system with consumption of chemoattractant in higher dimensions, J. Math. Phys., 58 (2017), 011503, 22 pp. doi: 10.1063/1.4974245.

[8]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.

[9]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super Pisa Cl. Sci., 24 (1997), 633-683. 

[10]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.

[11]

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.

[12]

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

[13]

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.

[14]

N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint.

[15]

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

[16]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. 

[17]

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. 

[18]

L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737. 

[19]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[20]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart, 10 (2002), 501-543. 

[21]

T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Anal., (2006), Art. ID 23061, 21 pp. doi: 10.1155/AAA/2006/23061.

[22]

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.

[23]

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

[24]

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.

[25]

M. Winkler, A critical exponent in a degenerate parabolic parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.  doi: 10.1002/mma.319.

[26]

M. Winkler, Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.  doi: 10.1088/1361-6544/aa565b.

[27]

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.

[28]

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.

[29]

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

[30]

H. Yu, W. Wang and S. Zheng, Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1635-1644.  doi: 10.3934/dcdsb.2017078.

show all references

References:
[1]

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.

[2]

T. Cieślak and C. Morales-Rodrigo, Quasilinear non-uniformly parabolic-elliptic system modelling chemotaxis with volume filling effect: Existence and uniqueness of global-in-time solutions, Topol. Methods Nonlinear Anal., 29 (2007), 361-381. 

[3]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.

[4]

T. Cieślak and M. Winkler, Global bounded solutions in a two-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal. Real World Appl., 35 (2017), 1-19.  doi: 10.1016/j.nonrwa.2016.10.002.

[5]

T. Cieślak and M. Winkler, Stabilization in a higher-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal., 159 (2017), 129-144.  doi: 10.1016/j.na.2016.04.013.

[6]

M. Ding and S. Zheng, $ L^\gamma $-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity, Discrete Contin. Dyn. Syst. Ser. B, Online First, (2018). doi: 10.3934/dcdsb.2018295.

[7]

L. Fan and H. Jin, Global existence and asymptotic behavior to a chemotaxis system with consumption of chemoattractant in higher dimensions, J. Math. Phys., 58 (2017), 011503, 22 pp. doi: 10.1063/1.4974245.

[8]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.

[9]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super Pisa Cl. Sci., 24 (1997), 633-683. 

[10]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.

[11]

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.

[12]

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

[13]

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.

[14]

N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint.

[15]

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

[16]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. 

[17]

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. 

[18]

L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737. 

[19]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[20]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart, 10 (2002), 501-543. 

[21]

T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Anal., (2006), Art. ID 23061, 21 pp. doi: 10.1155/AAA/2006/23061.

[22]

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.

[23]

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

[24]

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.

[25]

M. Winkler, A critical exponent in a degenerate parabolic parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.  doi: 10.1002/mma.319.

[26]

M. Winkler, Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.  doi: 10.1088/1361-6544/aa565b.

[27]

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.

[28]

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.

[29]

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

[30]

H. Yu, W. Wang and S. Zheng, Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1635-1644.  doi: 10.3934/dcdsb.2017078.

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