March  2014, 7(1): 9-28. doi: 10.3934/krm.2014.7.9

Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$

1. 

Department of Mathematics, Ocean University of China, Qingdao, 266003, China

2. 

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

3. 

Department of Mathematical Sciences, Peking University, Beijing, 100871, China

Received  October 2013 Revised  October 2013 Published  December 2013

This paper establishes the hyper-contractivity in $L^\infty(\mathbb{R}^d)$ (it's known as ultra-contractivity) for the multi-dimensional Keller-Segel systems with the diffusion exponent $m>1-2/d$. The results show that for the supercritical and critical case $1-2/d < m ≤ 2-2/d$, if $||U_0||_{d(2-m)/2} < C_{d,m}$ where $C_{d,m}$ is a universal constant, then for any $t>0$, $||u(\cdot,t)||_{L^\infty(\mathbb{R}^d)}$ is bounded and decays as $t$ goes to infinity. For the subcritical case $m>2-2/d$, the solution $u(\cdot,t) \in L^\infty(\mathbb{R}^d)$ with any initial data $U_0 \in L_+^1(\mathbb{R}^d)$ for any positive time.
Citation: Shen Bian, Jian-Guo Liu, Chen Zou. Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$. Kinetic and Related Models, 2014, 7 (1) : 9-28. doi: 10.3934/krm.2014.7.9
References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[2]

J. Bedrossian, Intermediate asymptotics for critical and supercritical aggregation equations and Patlak-Keller-Segel models, Comm. Math. Sci., 9 (2011), 1143-1161. doi: 10.4310/CMS.2011.v9.n4.a11.

[3]

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

[4]

A. Blanchet, J. A. 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.

[5]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Eletron. J. Differ. Equ., 2006, 32 pp. (electronic).

[6]

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

[7]

V. Calvez, L. Corrias and M. A. Ebde, Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension, Comm. Part. Diff. Eq., 37 (2012), 561-584. doi: 10.1080/03605302.2012.655824.

[8]

E. A. Carlen, J. A. Carrillo and M. Loss, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows, Proc. Nat. Acad. USA, 107 (2010), 19696-19701. doi: 10.1073/pnas.1008323107.

[9]

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

[10]

M. Del Pino, J. Dolbeault and I. Gentil, Nonlinear diffusions, hypercontractivity and the optimal $L^p$-Euclidean logarithmic Sobolev inequality, J. Math. Anal. Appl., 293 (2004), 375-388. doi: 10.1016/j.jmaa.2003.10.009.

[11]

M. Herrero, E. Medina and 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.

[12]

M. Herrero, E. Medina and J. L. Velázquez, Self-similar blow-up for a reaction-diffusion system, J. Comp. Appl. Math., 97 (1998), 99-119. doi: 10.1016/S0377-0427(98)00104-6.

[13]

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

[14]

I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2012), 568-602. doi: 10.1137/110823584.

[15]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics. V. 14, American Mathematical Society Providence, Rhode Island, 2nd edition, 2001. doi: 10.1080/13683500108667891.

[16]

B. Perthame, Transport Equations in Biology, Birkhaeuser Verlag, Basel-Boston-Berlin, 2007.

[17]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate keller-segel systems, Diff. Int. Eqns., 19 (2006), 841-876.

[18]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate keller-segel model with a power factor in drift term, J. Diff. Eqns., 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003.

[19]

J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford Lecture Ser. Math. Appl., vol. 33, 2006.

[20]

J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

show all references

References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[2]

J. Bedrossian, Intermediate asymptotics for critical and supercritical aggregation equations and Patlak-Keller-Segel models, Comm. Math. Sci., 9 (2011), 1143-1161. doi: 10.4310/CMS.2011.v9.n4.a11.

[3]

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

[4]

A. Blanchet, J. A. 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.

[5]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Eletron. J. Differ. Equ., 2006, 32 pp. (electronic).

[6]

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

[7]

V. Calvez, L. Corrias and M. A. Ebde, Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension, Comm. Part. Diff. Eq., 37 (2012), 561-584. doi: 10.1080/03605302.2012.655824.

[8]

E. A. Carlen, J. A. Carrillo and M. Loss, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows, Proc. Nat. Acad. USA, 107 (2010), 19696-19701. doi: 10.1073/pnas.1008323107.

[9]

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

[10]

M. Del Pino, J. Dolbeault and I. Gentil, Nonlinear diffusions, hypercontractivity and the optimal $L^p$-Euclidean logarithmic Sobolev inequality, J. Math. Anal. Appl., 293 (2004), 375-388. doi: 10.1016/j.jmaa.2003.10.009.

[11]

M. Herrero, E. Medina and 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.

[12]

M. Herrero, E. Medina and J. L. Velázquez, Self-similar blow-up for a reaction-diffusion system, J. Comp. Appl. Math., 97 (1998), 99-119. doi: 10.1016/S0377-0427(98)00104-6.

[13]

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

[14]

I. Kim and Y. Yao, The Patlak-Keller-Segel model and its variations: properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2012), 568-602. doi: 10.1137/110823584.

[15]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics. V. 14, American Mathematical Society Providence, Rhode Island, 2nd edition, 2001. doi: 10.1080/13683500108667891.

[16]

B. Perthame, Transport Equations in Biology, Birkhaeuser Verlag, Basel-Boston-Berlin, 2007.

[17]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate keller-segel systems, Diff. Int. Eqns., 19 (2006), 841-876.

[18]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate keller-segel model with a power factor in drift term, J. Diff. Eqns., 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003.

[19]

J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford Lecture Ser. Math. Appl., vol. 33, 2006.

[20]

J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

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