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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.   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.   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).   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).   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.   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. Poincar\'e Anal. Non Lin\'eaire, 25 (2008), 105.   Google Scholar

[8]

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

[10]

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

[15]

E. H. Lieb and M. Loss, "Analysis,'', Graduate Studies in Mathematics \textbf{14}, 14 (2001).   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[22]

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

[23]

D. Slepčev and M. C. Pugh, Selfsimilar blowup of unstable thin-film equations,, Indiana Univ. Math. J., 54 (2005), 1697.   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.   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.   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.   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.   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.   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.   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).   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).   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.   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. Poincar\'e Anal. Non Lin\'eaire, 25 (2008), 105.   Google Scholar

[8]

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

[10]

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

[15]

E. H. Lieb and M. Loss, "Analysis,'', Graduate Studies in Mathematics \textbf{14}, 14 (2001).   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[22]

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

[23]

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

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