Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity

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January 2014, 19(1): 231-256. doi: 10.3934/dcdsb.2014.19.231

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

Received December 2012 Revised August 2013 Published December 2013

Abstract
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We study a one-dimensional parabolic PDE with degenerate diffusion and non-Lipschitz nonlinearity involving the derivative. This evolution equation arises when searching radially symmetric solutions of a chemotaxis model of Patlak-Keller-Segel type. We prove its local in time wellposedness in some appropriate space, a blow-up alternative, regularity results and give an idea of the shape of solutions. A transformed and an approximate problem naturally appear in the way of the proof and are also crucial in [22] in order to study the global behaviour of solutions of the equation for a critical parameter, more precisely to show the existence of a critical mass.

Keywords: blow-up criterion., parabolic, Keller-Segel, system, degenerate, wellposedness, regularity, Chemotaxis.

Mathematics Subject Classification: 35A01, 35A02, 35A09, 35B44, 35K40, 35K51, 35K65, 35Q9.

Citation: Alexandre Montaru. Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 231-256. doi: 10.3934/dcdsb.2014.19.231

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.
[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. 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. 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. 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.
[6]	V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbbR^2$,, Commun. MAth. Sci., 6 (2008), 417. 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. 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. doi: 10.1016/j.crma.2004.08.011.
[9]	A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall, (1964).
[10]	T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and its Applications, (1998).
[11]	M. A. Herrero, The mathematics of chemotaxis, handbook of differential equations: Evolutionary equations,, Vol. III, III (2007), 137. 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. 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. 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.
[15]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II,, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51.
[16]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, Journal of Differential Equations, 215 (2005), 52. 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. doi: 10.1007/s00285-008-0201-3.
[18]	B. Hu, Blow-up Theories for Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (2018). 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 (): 1852. 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. 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, (1995). 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 , (): 0951. 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. doi: 10.1007/BF02476407.
[24]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). 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. doi: 10.1007/s10492-004-6431-9.
[26]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,, Springer-Verlag, (1984). 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, (2007).

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.
[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. 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. 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. 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.
[6]	V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbbR^2$,, Commun. MAth. Sci., 6 (2008), 417. 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. 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. doi: 10.1016/j.crma.2004.08.011.
[9]	A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall, (1964).
[10]	T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and its Applications, (1998).
[11]	M. A. Herrero, The mathematics of chemotaxis, handbook of differential equations: Evolutionary equations,, Vol. III, III (2007), 137. 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. 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. 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.
[15]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II,, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51.
[16]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, Journal of Differential Equations, 215 (2005), 52. 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. doi: 10.1007/s00285-008-0201-3.
[18]	B. Hu, Blow-up Theories for Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (2018). 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 (): 1852. 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. 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, (1995). 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 , (): 0951. 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. doi: 10.1007/BF02476407.
[24]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). 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. doi: 10.1007/s10492-004-6431-9.
[26]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations,, Springer-Verlag, (1984). 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, (2007).

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