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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 |
References:
[1] |
T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry, 11 (1976), 573-598. |
[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. |
[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. |
[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. |
[5] |
F. E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874. |
[6] |
F. E. Browder, Non-linear equations of evolution, Ann. of Math., 80 (1964), 485-523.
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), 1077-1102.
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), 103-120. |
[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. |
[10] |
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 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), 225-290.
doi: 10.1007/BF00400111. |
[12] |
P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math., 115 (1966), 271-310.
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), 103-165. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
B. Perthame, Transport Equations in Biology, Birkhäuser Verlag, Basel, 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), 133-180. |
[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. |
[26] |
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.
doi: 10.1007/BF02418013. |
[27] |
J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, The Clarendon Press, Oxford University Press, Oxford, 2007. |
[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. |
show all references
References:
[1] |
T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry, 11 (1976), 573-598. |
[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. |
[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. |
[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. |
[5] |
F. E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874. |
[6] |
F. E. Browder, Non-linear equations of evolution, Ann. of Math., 80 (1964), 485-523.
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), 1077-1102.
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), 103-120. |
[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. |
[10] |
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 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), 225-290.
doi: 10.1007/BF00400111. |
[12] |
P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math., 115 (1966), 271-310.
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), 103-165. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
B. Perthame, Transport Equations in Biology, Birkhäuser Verlag, Basel, 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), 133-180. |
[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. |
[26] |
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.
doi: 10.1007/BF02418013. |
[27] |
J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, The Clarendon Press, Oxford University Press, Oxford, 2007. |
[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. |
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