doi: 10.3934/dcdsb.2019059

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

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  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.

Citation: Mengyao Ding, Xiangdong Zhao. $ L^\sigma $-measure criteria for boundedness in a quasilinear parabolic-parabolic Keller-Segel system with supercritical sensitivity. Discrete & Continuous Dynamical Systems - B, 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. IshidaK. 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. NagaiT. 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. YuW. 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. YuW. 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. IshidaK. 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. NagaiT. 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. YuW. 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. YuW. 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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