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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

Alexandre Montaru 1,

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

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), 133-147.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbbR^2$, Commun. MAth. Sci., 6 (2008), 417-447. doi: 10.4310/CMS.2008.v6.n2.a8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 xiv+347 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. A. Herrero and J. L. Velazquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623. doi: 10.1007/BF01445268.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbbR^2$, Commun. MAth. Sci., 6 (2008), 417-447. doi: 10.4310/CMS.2008.v6.n2.a8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 xiv+347 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. A. Herrero and J. L. Velazquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623. doi: 10.1007/BF01445268.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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