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March  2017, 22(2): 307-338. doi: 10.3934/dcdsb.2017015

Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model

Wenting Cong 1,2,, and  Jian-Guo Liu 2,

1. 

School of Mathematics, Jilin University, Changchun 130012, China
2. 

Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA

* Corresponding author: Wenting Cong

Received  February 2016 Revised  October 2016 Published  December 2016

Fund Project: The first author is supported by NSFC grant 11271154.

This paper investigates the existence of a uniform in time $L^{∞}$ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent $0<m<2-\frac{2}{d}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(2-m)}{2}}$ norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution $u(x,t)$ satisfies mass conservation when $m>1-\frac{2}{d}$. We also prove the local existence of weak entropy solutions and a blow-up criterion for general $L^1\cap L^{∞}$ initial data.

Keywords: Chemotaxis, fast diffusion, critical space, semi-group theory, global existence.
Mathematics Subject Classification: Primary: 35K65, 35K59, 92C1.
Citation: Wenting Cong, Jian-Guo Liu. Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 307-338. doi: 10.3934/dcdsb.2017015
References:
[1]

S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m>0$, Comm. Math. Phys., 323 (2013), 1017-1070.  doi: 10.1007/s00220-013-1777-z.  Google Scholar

[2]

S. Bian, J.-G. Liu and C. Zou, Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$, Kinet. Relat. Models, 7 (2014), 9-28.  doi: 10.3934/krm.2014.7.9.  Google Scholar

[3]

A. Blanchet, J. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in ${\mathbb{R}}^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481.  doi: 10.1002/cpa.20225.  Google Scholar

[4]

M. Brenner, P. Constantin, L. Kadanoff, A. Schenkel and S. Venkataramani, Diffusion, attraction and collapse, Nonlinearity, 12 (1999), 1071-1098.  doi: 10.1088/0951-7715/12/4/320.  Google Scholar

[5]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in ${\mathbb{R}}^2$, Commun. Math. Sci., 6 (2008), 417-447.  doi: 10.4310/CMS.2008.v6.n2.a8.  Google Scholar

[6]

X. Chen, A. Jüngel and J.-G. Liu, A note on Aubin-Lions-Dubinskiĭ lemmas, Acta Appl. Math., 133 (2014), 33-43.  doi: 10.1007/s10440-013-9858-8.  Google Scholar

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L. Corrias and B. Perthame, Critical space for the parabolic-parabolic Keller-Segel model in ${\mathbb{R}}^d$, C. R. Math. Acad. Sci. Paris, 342 (2006), 745-750.  doi: 10.1016/j.crma.2006.03.008.  Google Scholar

[8]

L. Evans, Partial Differential Equations Second edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. Google Scholar

[9]

M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.  doi: 10.1088/0951-7715/10/6/016.  Google Scholar

[10]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.  Google Scholar

[11]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.   Google Scholar

[12]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 2469-2491.  doi: 10.1016/j.jde.2011.08.047.  Google Scholar

[13]

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

[14]

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

[15]

P. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^{∞}$-functional calculus, Lecture Notes in Math., 1855 (2004), 65-311.  doi: 10.1007/978-3-540-44653-8_2.  Google Scholar

[16]

J.-G. Liu and J. Wang, A note on $L^{∞}$-bound and uniqueness to a degenerate Keller-Segel model, Acta Appl. Math., 142 (2016), 173-188.  doi: 10.1007/s10440-015-0022-5.  Google Scholar

[17]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. Google Scholar

[18]

B. Perthame, Transport Equations in Biology Birkhäuser Verlag, Basel, 2007. Google Scholar

[19]

Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180.   Google Scholar

[20]

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

[21]

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.  Google Scholar

show all references

References:
[1]

S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m>0$, Comm. Math. Phys., 323 (2013), 1017-1070.  doi: 10.1007/s00220-013-1777-z.  Google Scholar

[2]

S. Bian, J.-G. Liu and C. Zou, Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$, Kinet. Relat. Models, 7 (2014), 9-28.  doi: 10.3934/krm.2014.7.9.  Google Scholar

[3]

A. Blanchet, J. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in ${\mathbb{R}}^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481.  doi: 10.1002/cpa.20225.  Google Scholar

[4]

M. Brenner, P. Constantin, L. Kadanoff, A. Schenkel and S. Venkataramani, Diffusion, attraction and collapse, Nonlinearity, 12 (1999), 1071-1098.  doi: 10.1088/0951-7715/12/4/320.  Google Scholar

[5]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in ${\mathbb{R}}^2$, Commun. Math. Sci., 6 (2008), 417-447.  doi: 10.4310/CMS.2008.v6.n2.a8.  Google Scholar

[6]

X. Chen, A. Jüngel and J.-G. Liu, A note on Aubin-Lions-Dubinskiĭ lemmas, Acta Appl. Math., 133 (2014), 33-43.  doi: 10.1007/s10440-013-9858-8.  Google Scholar

[7]

L. Corrias and B. Perthame, Critical space for the parabolic-parabolic Keller-Segel model in ${\mathbb{R}}^d$, C. R. Math. Acad. Sci. Paris, 342 (2006), 745-750.  doi: 10.1016/j.crma.2006.03.008.  Google Scholar

[8]

L. Evans, Partial Differential Equations Second edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. Google Scholar

[9]

M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.  doi: 10.1088/0951-7715/10/6/016.  Google Scholar

[10]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.  Google Scholar

[11]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.   Google Scholar

[12]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 2469-2491.  doi: 10.1016/j.jde.2011.08.047.  Google Scholar

[13]

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

[14]

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

[15]

P. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^{∞}$-functional calculus, Lecture Notes in Math., 1855 (2004), 65-311.  doi: 10.1007/978-3-540-44653-8_2.  Google Scholar

[16]

J.-G. Liu and J. Wang, A note on $L^{∞}$-bound and uniqueness to a degenerate Keller-Segel model, Acta Appl. Math., 142 (2016), 173-188.  doi: 10.1007/s10440-015-0022-5.  Google Scholar

[17]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. Google Scholar

[18]

B. Perthame, Transport Equations in Biology Birkhäuser Verlag, Basel, 2007. Google Scholar

[19]

Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180.   Google Scholar

[20]

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

[21]

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.  Google Scholar

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