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Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity
1. | Département de Mathématiques - Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse, France |
References:
[1] |
P. Biler, G. Karch, P. Laurençot and T. Nadzieja, The $8\pi$-problem for radially symmetric solutions of a chemotaxis model in a disk, Topol. Methods Nonlinear Anal., 27 (2006), 133-147. |
[2] |
P. Biler, G. Karch, P. Laurençot and T. Nadzieja, The 8$\pi$-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Methods Appl. Sci., 29 (2006), 1563-1583.
doi: 10.1002/mma.743. |
[3] |
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. |
[4] |
A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical two-dimensional Patlak-Keller-Segel model, 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: Opti- mal critical mass and qualitative properties of solutions, Electron. J. Differential Equations, 44 (2006), 1-32. |
[6] |
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. |
[7] |
K. Djie and M. Winkler, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA, 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[8] |
J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in R2, C. R. Math. Acad. Sci. Paris, 339 (2004), 611-616.
doi: 10.1016/j.crma.2004.08.011. |
[9] |
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 xiv+347 pp. |
[10] |
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. Clarendon Press, Oxford University Press, New York, 1998. xiv+186 pp. |
[11] |
M. A. Herrero, The mathematics of chemotaxis, handbook of differential equations: Evolutionary equations, Vol. III, 137-193, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007.
doi: 10.1016/S1874-5717(07)80005-3. |
[12] |
M. A. Herrero and L. Sastre, Models of aggregation in dictyostelium discoideum: On the track of spiral waves, Networks and Heterogeneous Media, 1 (2006), 241-258.
doi: 10.3934/nhm.2006.1.241. |
[13] |
M. A. Herrero and J. L. Velazquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.
doi: 10.1007/BF01445268. |
[14] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[15] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69. |
[16] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, Journal of Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[17] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[18] |
B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Lecture Notes in Mathematics, 2018. Springer, Heidelberg, 2011. x+125 pp.
doi: 10.1007/978-3-642-18460-4. |
[19] |
N. I. Kavallaris and P. Souplet, Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk, SIAM J. Math. Anal., 40 (2008/09), 1852-1881.
doi: 10.1137/080722229. |
[20] |
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. |
[21] |
A. Lunard, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995. xviii+424 pp.
doi: 10.1007/978-3-0348-9234-6. |
[22] |
A. Montaru, A semilinear parabolic-elliptic chemotaxis system with critical mass in any space dimension, (accepted in Nonlinearity http://iopscience.iop.org/0951-7715/26/9/2669/).
doi: 10.1088/0951-7715/26/9/2669. |
[23] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[24] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279 pp.
doi: 10.1007/978-1-4612-5561-1. |
[25] |
B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564.
doi: 10.1007/s10492-004-6431-9. |
[26] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. x+261 pp.
doi: 10.1007/978-1-4612-5282-5. |
[27] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts. Birkhäuser Verlag, Basel, 2007. xii+584 pp. |
show all references
References:
[1] |
P. Biler, G. Karch, P. Laurençot and T. Nadzieja, The $8\pi$-problem for radially symmetric solutions of a chemotaxis model in a disk, Topol. Methods Nonlinear Anal., 27 (2006), 133-147. |
[2] |
P. Biler, G. Karch, P. Laurençot and T. Nadzieja, The 8$\pi$-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Methods Appl. Sci., 29 (2006), 1563-1583.
doi: 10.1002/mma.743. |
[3] |
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. |
[4] |
A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical two-dimensional Patlak-Keller-Segel model, 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: Opti- mal critical mass and qualitative properties of solutions, Electron. J. Differential Equations, 44 (2006), 1-32. |
[6] |
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. |
[7] |
K. Djie and M. Winkler, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA, 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[8] |
J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in R2, C. R. Math. Acad. Sci. Paris, 339 (2004), 611-616.
doi: 10.1016/j.crma.2004.08.011. |
[9] |
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 xiv+347 pp. |
[10] |
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. Clarendon Press, Oxford University Press, New York, 1998. xiv+186 pp. |
[11] |
M. A. Herrero, The mathematics of chemotaxis, handbook of differential equations: Evolutionary equations, Vol. III, 137-193, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007.
doi: 10.1016/S1874-5717(07)80005-3. |
[12] |
M. A. Herrero and L. Sastre, Models of aggregation in dictyostelium discoideum: On the track of spiral waves, Networks and Heterogeneous Media, 1 (2006), 241-258.
doi: 10.3934/nhm.2006.1.241. |
[13] |
M. A. Herrero and J. L. Velazquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.
doi: 10.1007/BF01445268. |
[14] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[15] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69. |
[16] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, Journal of Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[17] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[18] |
B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Lecture Notes in Mathematics, 2018. Springer, Heidelberg, 2011. x+125 pp.
doi: 10.1007/978-3-642-18460-4. |
[19] |
N. I. Kavallaris and P. Souplet, Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk, SIAM J. Math. Anal., 40 (2008/09), 1852-1881.
doi: 10.1137/080722229. |
[20] |
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. |
[21] |
A. Lunard, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995. xviii+424 pp.
doi: 10.1007/978-3-0348-9234-6. |
[22] |
A. Montaru, A semilinear parabolic-elliptic chemotaxis system with critical mass in any space dimension, (accepted in Nonlinearity http://iopscience.iop.org/0951-7715/26/9/2669/).
doi: 10.1088/0951-7715/26/9/2669. |
[23] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[24] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279 pp.
doi: 10.1007/978-1-4612-5561-1. |
[25] |
B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564.
doi: 10.1007/s10492-004-6431-9. |
[26] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. x+261 pp.
doi: 10.1007/978-1-4612-5282-5. |
[27] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts. Birkhäuser Verlag, Basel, 2007. xii+584 pp. |
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