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

A degenerate $p$-Laplacian Keller-Segel model

Wenting Cong 1, 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

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.
Keywords: fast diffusion, monotone operator, global existence, Chemotaxis, critical space, non-Newtonian filtration..
Mathematics Subject Classification: Primary: 35K65, 35K92, 92C1.
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), 573-598.  Google Scholar

[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), 1017-1070. doi: 10.1007/s00220-013-1777-z.  Google Scholar

[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), 9-28. doi: 10.3934/krm.2014.7.9.  Google Scholar

[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), 133-168. doi: 10.1007/s00526-008-0200-7.  Google Scholar

[5]

F. E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874.  Google Scholar

[6]

F. E. Browder, Non-linear equations of evolution, Ann. of Math., 80 (1964), 485-523. doi: 10.2307/1970660.  Google Scholar

[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), 1077-1102. doi: 10.1137/110839102.  Google Scholar

[8]

L. Chen and J. Wang, Exact criterion for global existence and blow up to a degenerate Keller-Segel system, Doc. Math., 19 (2014), 103-120.  Google Scholar

[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), 493-516. doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[10]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[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), 225-290. doi: 10.1007/BF00400111.  Google Scholar

[12]

P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math., 115 (1966), 271-310. doi: 10.1007/BF02392210.  Google Scholar

[13]

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

[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]

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.  Google Scholar

[16]

O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva and S. Smith, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968.  Google Scholar

[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), 97-107.  Google Scholar

[18]

E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[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), 173-188. doi: 10.1007/s10440-015-0022-5.  Google Scholar

[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), 1038-1041. doi: 10.1073/pnas.50.6.1038.  Google Scholar

[21]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346. doi: 10.1215/S0012-7094-62-02933-2.  Google Scholar

[22]

B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564. doi: 10.1007/s10492-004-6431-9.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.  Google Scholar

[27]

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

[28]

Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812799791.  Google Scholar

show all references

References:
[1]

T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry, 11 (1976), 573-598.  Google Scholar

[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), 1017-1070. doi: 10.1007/s00220-013-1777-z.  Google Scholar

[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), 9-28. doi: 10.3934/krm.2014.7.9.  Google Scholar

[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), 133-168. doi: 10.1007/s00526-008-0200-7.  Google Scholar

[5]

F. E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874.  Google Scholar

[6]

F. E. Browder, Non-linear equations of evolution, Ann. of Math., 80 (1964), 485-523. doi: 10.2307/1970660.  Google Scholar

[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), 1077-1102. doi: 10.1137/110839102.  Google Scholar

[8]

L. Chen and J. Wang, Exact criterion for global existence and blow up to a degenerate Keller-Segel system, Doc. Math., 19 (2014), 103-120.  Google Scholar

[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), 493-516. doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[10]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[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), 225-290. doi: 10.1007/BF00400111.  Google Scholar

[12]

P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math., 115 (1966), 271-310. doi: 10.1007/BF02392210.  Google Scholar

[13]

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

[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]

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.  Google Scholar

[16]

O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva and S. Smith, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968.  Google Scholar

[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), 97-107.  Google Scholar

[18]

E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[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), 173-188. doi: 10.1007/s10440-015-0022-5.  Google Scholar

[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), 1038-1041. doi: 10.1073/pnas.50.6.1038.  Google Scholar

[21]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346. doi: 10.1215/S0012-7094-62-02933-2.  Google Scholar

[22]

B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564. doi: 10.1007/s10492-004-6431-9.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.  Google Scholar

[27]

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

[28]

Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812799791.  Google Scholar

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