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December  2016, 9(4): 687-714. doi: 10.3934/krm.2016012

A degenerate $p$-Laplacian Keller-Segel model

1. 

School of Mathematics, Jilin University, Changchun 130012, China

2. 

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

Received  February 2016 Revised  April 2016 Published  September 2016

This paper investigates the existence of a uniform in time $L^{\infty}$ bounded weak solution for the $p$-Laplacian Keller-Segel system with the supercritical diffusion exponent $1 < p < \frac{3d}{d+1}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(3-p)}{p}}$ norm of initial data is smaller than a universal constant. We also prove the local existence of weak solutions and a blow-up criterion for general $L^1\cap L^{\infty}$ initial data.
Citation: Wenting Cong, Jian-Guo Liu. A degenerate $p$-Laplacian Keller-Segel model. Kinetic & Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012
References:
[1]

T. Aubin, Problèmes isopérimétriques et espaces de Sobolev,, J. Differential Geometry, 11 (1976), 573.

[2]

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. doi: 10.1007/s00220-013-1777-z.

[3]

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. doi: 10.3934/krm.2014.7.9.

[4]

A. Blanchet, J. A. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions,, Calc. Var. Partial Differential Equations, 35 (2009), 133. doi: 10.1007/s00526-008-0200-7.

[5]

F. E. Browder, Nonlinear elliptic boundary value problems,, Bull. Amer. Math. Soc., 69 (1963), 862.

[6]

F. E. Browder, Non-linear equations of evolution,, Ann. of Math., 80 (1964), 485. doi: 10.2307/1970660.

[7]

L. Chen, J.-G. Liu and J. Wang, Multidimensional degenerate Keller-Segel system with critical diffusion exponent $2n/(n+2)$,, SIAM J. Math. Anal., 44 (2012), 1077. doi: 10.1137/110839102.

[8]

L. Chen and J. Wang, Exact criterion for global existence and blow up to a degenerate Keller-Segel system,, Doc. Math., 19 (2014), 103.

[9]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results,, Ann. Inst. H. Poincaré Anal. Non Linéaire}, 15 (1998), 493. doi: 10.1016/S0294-1449(98)80032-2.

[10]

E. DiBenedetto, Degenerate Parabolic Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-0895-2.

[11]

E. DiBenedetto and M. A. Herrero, Non-negative solutions of the evolution $p$-Laplacian equation. Initial traces and Cauchy problem when $1 < p < 2$,, Arch. Rational Mech. Anal., 111 (1990), 225. doi: 10.1007/BF00400111.

[12]

P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations,, Acta Math., 115 (1966), 271. doi: 10.1007/BF02392210.

[13]

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

[14]

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

[15]

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. doi: 10.1137/110823584.

[16]

O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva and S. Smith, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society, (1968).

[17]

J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder,, Bull. Soc. Math. France, 93 (1965), 97.

[18]

E. H. Lieb and M. Loss, Analysis,, Second edition, (2001). doi: 10.1090/gsm/014.

[19]

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

[20]

G. J. Minty, On a monotonicity method for the solution of nonlinear equations in Banach spaces,, Proc. Nat. Acad. Sci. U.S.A., 50 (1963), 1038. doi: 10.1073/pnas.50.6.1038.

[21]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space,, Duke Math. J., 29 (1962), 341. doi: 10.1215/S0012-7094-62-02933-2.

[22]

B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539. doi: 10.1007/s10492-004-6431-9.

[23]

B. Perthame, Transport Equations in Biology,, Birkhäuser Verlag, (2007).

[24]

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

[25]

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

[26]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353. doi: 10.1007/BF02418013.

[27]

J. L. Vázquez, The Porous Medium Equation: Mathematical Theory,, The Clarendon Press, (2007).

[28]

Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations,, World Scientific Publishing Co., (2001). doi: 10.1142/9789812799791.

show all references

References:
[1]

T. Aubin, Problèmes isopérimétriques et espaces de Sobolev,, J. Differential Geometry, 11 (1976), 573.

[2]

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. doi: 10.1007/s00220-013-1777-z.

[3]

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. doi: 10.3934/krm.2014.7.9.

[4]

A. Blanchet, J. A. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions,, Calc. Var. Partial Differential Equations, 35 (2009), 133. doi: 10.1007/s00526-008-0200-7.

[5]

F. E. Browder, Nonlinear elliptic boundary value problems,, Bull. Amer. Math. Soc., 69 (1963), 862.

[6]

F. E. Browder, Non-linear equations of evolution,, Ann. of Math., 80 (1964), 485. doi: 10.2307/1970660.

[7]

L. Chen, J.-G. Liu and J. Wang, Multidimensional degenerate Keller-Segel system with critical diffusion exponent $2n/(n+2)$,, SIAM J. Math. Anal., 44 (2012), 1077. doi: 10.1137/110839102.

[8]

L. Chen and J. Wang, Exact criterion for global existence and blow up to a degenerate Keller-Segel system,, Doc. Math., 19 (2014), 103.

[9]

L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results,, Ann. Inst. H. Poincaré Anal. Non Linéaire}, 15 (1998), 493. doi: 10.1016/S0294-1449(98)80032-2.

[10]

E. DiBenedetto, Degenerate Parabolic Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-0895-2.

[11]

E. DiBenedetto and M. A. Herrero, Non-negative solutions of the evolution $p$-Laplacian equation. Initial traces and Cauchy problem when $1 < p < 2$,, Arch. Rational Mech. Anal., 111 (1990), 225. doi: 10.1007/BF00400111.

[12]

P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations,, Acta Math., 115 (1966), 271. doi: 10.1007/BF02392210.

[13]

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

[14]

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

[15]

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. doi: 10.1137/110823584.

[16]

O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva and S. Smith, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society, (1968).

[17]

J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder,, Bull. Soc. Math. France, 93 (1965), 97.

[18]

E. H. Lieb and M. Loss, Analysis,, Second edition, (2001). doi: 10.1090/gsm/014.

[19]

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

[20]

G. J. Minty, On a monotonicity method for the solution of nonlinear equations in Banach spaces,, Proc. Nat. Acad. Sci. U.S.A., 50 (1963), 1038. doi: 10.1073/pnas.50.6.1038.

[21]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space,, Duke Math. J., 29 (1962), 341. doi: 10.1215/S0012-7094-62-02933-2.

[22]

B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539. doi: 10.1007/s10492-004-6431-9.

[23]

B. Perthame, Transport Equations in Biology,, Birkhäuser Verlag, (2007).

[24]

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

[25]

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

[26]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353. doi: 10.1007/BF02418013.

[27]

J. L. Vázquez, The Porous Medium Equation: Mathematical Theory,, The Clarendon Press, (2007).

[28]

Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations,, World Scientific Publishing Co., (2001). doi: 10.1142/9789812799791.

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