July  2019, 24(7): 2971-2988. doi: 10.3934/dcdsb.2018295

$ L^γ$-measure criteria for boundedness in a quasilinear parabolic-elliptic 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  November 2017 Revised  July 2018 Published  October 2018

Fund Project: The first author is supported by the National Natural Science Foundation of China (11571020, 11671021, 11171048).

This paper studies the parabolic-elliptic Keller-Segel system with supercritical sensitivity: $u_{t} = \nabla·(D(u)\nabla u)-\nabla ·(S(u)\nabla v)$, $0 = Δ v -v+u$ in $Ω× (0,T)$, where the bounded domain $Ω\subset\mathbb{R}^n$, $n≥2$, subject to the non-flux boundary conditions, $D(u)≥ a_0(u+1)^{-q}$, $0≤ S(u)≤ b_0u(u+1)^{α-q-1}$ with $q \in \mathbb{R}$, $α>\frac{2}{n}$, and $a_0, b_0>0$. It is proved that the problem possesses a unique globally bounded solution for $α>\frac{2}{n}$ whenever $\|u_0\|_{L^{\frac{nα}{2}}}$ is sufficiently small. In addition, we establish the large-time behavior of solutions when $q = 0$.

Citation: Mengyao Ding, Sining Zheng. $ L^γ$-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2971-2988. doi: 10.3934/dcdsb.2018295
References:
[1]

M. BurgerM. Di Francesco and Y. Dolak-Struss, The Keller-Segel model for chemotaxis with prevention of overcrowding: Linear versus nonlinear diffusion, SIAM J. Math. Anal., 38 (2006), 1288-1315.  doi: 10.1137/050637923.  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

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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.  Google Scholar

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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.  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.   Google Scholar

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R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588.  doi: 10.1016/j.jmaa.2004.12.009.  Google Scholar

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

[13]

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

[14]

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

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

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Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.  Google Scholar

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

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M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071.  Google Scholar

[22]

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

[23]

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

[24]

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.  Google Scholar

[25]

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

[26]

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.  Google Scholar

show all references

References:
[1]

M. BurgerM. Di Francesco and Y. Dolak-Struss, The Keller-Segel model for chemotaxis with prevention of overcrowding: Linear versus nonlinear diffusion, SIAM J. Math. Anal., 38 (2006), 1288-1315.  doi: 10.1137/050637923.  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]

X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), Paper No. 107, 39pp. doi: 10.1007/s00526-016-1027-2.  Google Scholar

[4]

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.   Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.   Google Scholar

[11]

R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588.  doi: 10.1016/j.jmaa.2004.12.009.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Y. Sugiyama, Global existence and decay properties of solutions for some degenerate quasilinear parabolic systems modelling chemotaxis, Nonlinear Anal., 63 (2005), 1051-1062.   Google Scholar

[17]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.  Google Scholar

[18]

Y. Sugiyama and Y. Yahagi, Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type, J. Differential Equations, 250 (2011), 3047-3087.  doi: 10.1016/j.jde.2011.01.016.  Google Scholar

[19]

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

[20]

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

[21]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071.  Google Scholar

[22]

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

[23]

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

[24]

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.  Google Scholar

[25]

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

[26]

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.  Google Scholar

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