American Institute of Mathematical Sciences

Advanced
January  2012, 11(1): 47-60. doi: 10.3934/cpaa.2012.11.47

Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion

Adrien Blanchet 1, and  Philippe Laurençot 2,

1. 

GREMAQ, CNRS UMR 5604, INRA UMR 1291, Université de Toulouse, 21 Allée de Brienne, F--31000 Toulouse, France
2. 

Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse, F–31062 Toulouse cedex 9

Received  November 2009 Revised  October 2010 Published  September 2011

For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is a critical mass $M_c>0$ such that all solutions with initial data of mass smaller or equal to $M_c$ exist globally while the solution blows up in finite time for a large class of initial data with mass greater than $M_c$. Unlike in space dimension $2$, finite mass self-similar blowing-up solutions are shown to exist in space dimension $d\geq 3$.
Keywords: Backward self-similar solutions, Patlak-Keller-Segel model, chemotaxis, degenerate diffusion., blowup.
Mathematics Subject Classification: Primary: 35K65, 34C10; Secondary: 92C1.
Citation: Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
References:
[1]

A. Blanchet, E. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model,, preprint, ().   Google Scholar

[2]

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.  Google Scholar

[3]

A. Blanchet, J. A. Carrillo and Ph. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.  Google Scholar

[4]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 32 pp. (electronic).  Google Scholar

[5]

P.-H. Chavanis and C. Sire, Anomalous diffusion and collapse of self-gravitating Langevin particles in $D$ dimensions, Phys. Rev. E, 69 (2004), 016116. Google Scholar

[6]

J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in $R^2$, C. R. Math. Acad. Sci. Paris, 339 (2004), 611-616.  Google Scholar

[7]

P. L. Felmer, A. Quaas, M. Tang and J. Yu, Monotonicity properties for ground states of the scalar field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 105-119.  Google Scholar

[8]

R. H. Fowler, Further studies of Emden's and similar differential equations, Quart. J. Math., 2 (1931), 259-288. Google Scholar

[9]

M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.  Google Scholar

[10]

M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.  Google Scholar

[11]

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

[12]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  Google Scholar

[13]

E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. Google Scholar

[14]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $R^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  Google Scholar

[15]

E. H. Lieb and M. Loss, "Analysis,'' Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, second ed., 2001.  Google Scholar

[16]

P. M. Lushnikov, Critical chemotactic collapse, Phys. Lett. A, 374 (2010), 1678-1685. Google Scholar

[17]

T. Nagai, Behavior of solutions to a parabolic-elliptic system modelling chemotaxis, J. Korean Math. Soc., 37 (2000), 721-733.  Google Scholar

[18]

Y. Naito and T. Suzuki, Self-similarity in chemotaxis systems, Colloq. Math., 111 (2008), 11-34.  Google Scholar

[19]

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

[20]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.  Google Scholar

[21]

C. Sire and P.-H. Chavanis, Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions, Phys. Rev. E, 66 (2002), 046133.  Google Scholar

[22]

C. Sire and P.-H. Chavanis, Critical dynamics of self-gravitating Langevin particles and bacterial populations, Phys. Rev. E, 78 (2008), 061111.  Google Scholar

[23]

D. Slepčev and M. C. Pugh, Selfsimilar blowup of unstable thin-film equations, Indiana Univ. Math. J., 54 (2005), 1697-1738.  Google Scholar

[24]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.  Google Scholar

[25]

Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144.  Google Scholar

[26]

T. Suzuki and R. Takahashi, Degenerate parabolic equation with critical exponent derived from the kinetic theory, I, generation of the weak solution, Adv. Differential Equations, 14 (2009), 433-476.  Google Scholar

[27]

M. Tang, Uniqueness of positive radial solutions for $\Delta u - u + u^p=0$ on an annulus, J. Differential Equations, 189 (2003), 148-160.  Google Scholar

show all references

References:
[1]

A. Blanchet, E. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model,, preprint, ().   Google Scholar

[2]

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.  Google Scholar

[3]

A. Blanchet, J. A. Carrillo and Ph. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.  Google Scholar

[4]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), 32 pp. (electronic).  Google Scholar

[5]

P.-H. Chavanis and C. Sire, Anomalous diffusion and collapse of self-gravitating Langevin particles in $D$ dimensions, Phys. Rev. E, 69 (2004), 016116. Google Scholar

[6]

J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in $R^2$, C. R. Math. Acad. Sci. Paris, 339 (2004), 611-616.  Google Scholar

[7]

P. L. Felmer, A. Quaas, M. Tang and J. Yu, Monotonicity properties for ground states of the scalar field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 105-119.  Google Scholar

[8]

R. H. Fowler, Further studies of Emden's and similar differential equations, Quart. J. Math., 2 (1931), 259-288. Google Scholar

[9]

M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.  Google Scholar

[10]

M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.  Google Scholar

[11]

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

[12]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  Google Scholar

[13]

E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. Google Scholar

[14]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $R^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  Google Scholar

[15]

E. H. Lieb and M. Loss, "Analysis,'' Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, second ed., 2001.  Google Scholar

[16]

P. M. Lushnikov, Critical chemotactic collapse, Phys. Lett. A, 374 (2010), 1678-1685. Google Scholar

[17]

T. Nagai, Behavior of solutions to a parabolic-elliptic system modelling chemotaxis, J. Korean Math. Soc., 37 (2000), 721-733.  Google Scholar

[18]

Y. Naito and T. Suzuki, Self-similarity in chemotaxis systems, Colloq. Math., 111 (2008), 11-34.  Google Scholar

[19]

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

[20]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.  Google Scholar

[21]

C. Sire and P.-H. Chavanis, Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions, Phys. Rev. E, 66 (2002), 046133.  Google Scholar

[22]

C. Sire and P.-H. Chavanis, Critical dynamics of self-gravitating Langevin particles and bacterial populations, Phys. Rev. E, 78 (2008), 061111.  Google Scholar

[23]

D. Slepčev and M. C. Pugh, Selfsimilar blowup of unstable thin-film equations, Indiana Univ. Math. J., 54 (2005), 1697-1738.  Google Scholar

[24]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.  Google Scholar

[25]

Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144.  Google Scholar

[26]

T. Suzuki and R. Takahashi, Degenerate parabolic equation with critical exponent derived from the kinetic theory, I, generation of the weak solution, Adv. Differential Equations, 14 (2009), 433-476.  Google Scholar

[27]

M. Tang, Uniqueness of positive radial solutions for $\Delta u - u + u^p=0$ on an annulus, J. Differential Equations, 189 (2003), 148-160.  Google Scholar

[1]

Tian Xiang. On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4911-4946. doi: 10.3934/dcds.2014.34.4911
[2]

Jacob Bedrossian, Nancy Rodríguez. Inhomogeneous Patlak-Keller-Segel models and aggregation equations with nonlinear diffusion in $\mathbb{R}^d$. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1279-1309. doi: 10.3934/dcdsb.2014.19.1279
[3]

Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897
[4]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002
[5]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801
[6]

K. T. Joseph, Philippe G. LeFloch. Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits. Communications on Pure & Applied Analysis, 2002, 1 (1) : 51-76. doi: 10.3934/cpaa.2002.1.51
[7]

Jochen Merker, Aleš Matas. Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations. Conference Publications, 2015, 2015 (special) : 817-825. doi: 10.3934/proc.2015.0817
[8]

Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036
[9]

Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101
[10]

Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks & Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857
[11]

F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91
[12]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[13]

Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703
[14]

Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003
[15]

Wenting Cong, Jian-Guo Liu. A degenerate $p$-Laplacian Keller-Segel model. Kinetic & Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012
[16]

Kenneth H. Karlsen, Süleyman Ulusoy. On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion. Networks & Heterogeneous Media, 2016, 11 (1) : 181-201. doi: 10.3934/nhm.2016.11.181
[17]

Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046
[18]

Razvan Gabriel Iagar, Ana Isabel Muñoz, Ariel Sánchez. Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2022003
[19]

Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471
[20]

Meiyue Jiang, Juncheng Wei. $2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 785-803. doi: 10.3934/dcds.2016.36.785

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy