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

Previous Article
A dynamics approach to a low-order climate model
DCDS-B Home
This Issue
Next Article
Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent

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

Abstract
Related Papers

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 and 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.
[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.

[1]	Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[2]	Monica Marras, Stella Vernier Piro, Giuseppe Viglialoro. Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system. Conference Publications, 2015, 2015 (special) : 809-816. doi: 10.3934/proc.2015.0809
[3]	Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 233-255. doi: 10.3934/dcdss.2020013
[4]	Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure and Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243
[5]	Ansgar Jüngel, Oliver Leingang. Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4755-4782. doi: 10.3934/dcdsb.2019029
[6]	Jean Dolbeault, Christian Schmeiser. The two-dimensional Keller-Segel model after blow-up. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 109-121. doi: 10.3934/dcds.2009.25.109
[7]	Vincent Calvez, Thomas O. Gallouët. Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1175-1208. doi: 10.3934/dcds.2016.36.1175
[8]	Yuya Tanaka, Tomomi Yokota. Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022075
[9]	Sachiko Ishida, Tomomi Yokota. Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 211-232. doi: 10.3934/dcdss.2020012
[10]	Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231
[11]	Kenneth H. Karlsen, Süleyman Ulusoy. On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion. Networks and Heterogeneous Media, 2016, 11 (1) : 181-201. doi: 10.3934/nhm.2016.11.181
[12]	Jinhuan Wang, Li Chen, Liang Hong. Parabolic elliptic type Keller-Segel system on the whole space case. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1061-1084. doi: 10.3934/dcds.2016.36.1061
[13]	Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078
[14]	Wenting Cong, Jian-Guo Liu. A degenerate $p$-Laplacian Keller-Segel model. Kinetic and Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012
[15]	Wenting Cong, Jian-Guo Liu. Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 307-338. doi: 10.3934/dcdsb.2017015
[16]	Sachiko Ishida. $L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 335-344. doi: 10.3934/proc.2013.2013.335
[17]	Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure and Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339
[18]	Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure and Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287
[19]	Yoshifumi Mimura. Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1603-1630. doi: 10.3934/dcds.2017066
[20]	Kentarou Fujie, Chihiro Nishiyama, Tomomi Yokota. Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$. Conference Publications, 2015, 2015 (special) : 464-472. doi: 10.3934/proc.2015.0464

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Alexandre Montaru
on Google Scholar
Alexandre Montaru

[Back to Top]