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November  2015, 14(6): 2117-2126. doi: 10.3934/cpaa.2015.14.2117

Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane

Piotr Biler 1, , Ignacio Guerra 2, and  Grzegorz Karch 3,

1. 

Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50--384 Wrocław
2. 

Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile
3. 

Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław

Received  February 2014 Revised  September 2014 Published  September 2015

As it is well known, the parabolic-elliptic Keller-Segel system of chemotaxis on the plane has global-in-time regular nonnegative solutions with total mass below the critical value $8\pi$. Solutions with mass above $8\pi$ blow up in a finite time. We show that the case of the parabolic-parabolic Keller-Segel is different: each mass may lead to a global-in-time-solution, even if the initial data is a finite signed measure. These solutions need not be unique, even if we limit ourselves to nonnegative solutions.
Keywords: Chemotaxis, large global-in-time solutions., parabolic-parabolic Keller-Segel model.
Mathematics Subject Classification: Primary: 35Q92; Secondary: 35K4.
Citation: Piotr Biler, Ignacio Guerra, Grzegorz Karch. Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2117-2126. doi: 10.3934/cpaa.2015.14.2117
References:
[1]

J. Bedrossian and N. Masmoudi, Existence, uniqueness and Lipschitz dependence for Patlak-Keller-Segel and Navier-Stokes in $\mathbb R^2$ with measure-valued initial data,, Arch. Rational Mech. Anal., 214 (2014), 717. doi: 10.1007/s00205-014-0796-z. Google Scholar

[2]

P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation,, Studia Math., 114 (1995), 181. Google Scholar

[3]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 715. Google Scholar

[4]

P. Biler, Radially symmetric solutions of a chemotaxis model in the plane - the supercritical case, in: Parabolic and Navier-Stokes Equations,, Banach Center Publications, 81 (2008), 31. doi: 10.4064/bc81-0-2. Google Scholar

[5]

P. Biler and L. Brandolese, On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modelling chemotaxis,, Studia Math., 193 (2009), 241. doi: 10.4064/sm193-3-2. Google Scholar

[6]

P. Biler, L. Corrias and J. Dolbeault, Large mass self-similar solutions of the parabolic-parabolic Keller-Segel model,, J. Math. Biology, 63 (2011), 1. doi: 10.1007/s00285-010-0357-5. Google Scholar

[7]

P. Biler, G. Karch, Ph. Laurençot and T. Nadzieja, The $8\pi$-problem for radially symmetric solutions of a chemotaxis model in the plane,, Math. Methods in the Applied Sci., 29 (2006), 1563. doi: 10.1002/mma.743. Google Scholar

[8]

A. Blanchet, E. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model,, J. Functional Anal., 262 (2012), 2142. doi: 10.1016/j.jfa.2011.12.012. Google Scholar

[9]

A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbb R^2$,, Comm. Pure Appl. Math., 61 (2008), 1449. doi: 10.1002/cpa.20225. Google Scholar

[10]

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

[11]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbb R^2$,, Commun. Math. Sci., 6 (2008), 417. Google Scholar

[12]

L. Corrias, M. Escobedo and J. Matos, Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane,, J. Differential Equations, 257 (2014), 1840. doi: 10.1016/j.jde.2014.05.019. Google Scholar

[13]

J. Dolbeault and Ch. Schmeiser, The two-dimensional Keller-Segel model after blow-up,, Discrete Contin. Dyn. Syst., 25 (2009), 109. doi: 10.3934/dcds.2009.25.109. Google Scholar

[14]

Y. Giga, T. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity,, Arch. Rational Mech. Anal., 104 (1988), 223. doi: 10.1007/BF00281355. Google Scholar

[15]

H. Kozono and Y. Sugiyama, Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system,, J. Evol. Equ., 8 (2008), 353. doi: 10.1007/s00028-008-0375-6. Google Scholar

[16]

P.-G. Lemarié-Rieusset, Small data in an optimal Banach space for the parabolic-parabolic and parabolic-elliptic Keller-Segel equations in the whole space,, Adv. Diff. Eq., 18 (2013), 1189. Google Scholar

[17]

S. Luckhaus, Y. Sugiyama and J. J. L. Vélazquez, Measure valued solutions of the 2D Keller-Segel system,, Arch. Rational Mech. Anal., 206 (2012), 31. doi: 10.1007/s00205-012-0549-9. Google Scholar

[18]

N. Mizoguchi, Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane,, Calc. Var., 48 (2013), 491. doi: 10.1007/s00526-012-0558-4. Google Scholar

[19]

N. Mizoguchi and M. Winkler, (2013),, personal communication., (). Google Scholar

[20]

Y. Naito, Asymptotically self-similar solutions for the parabolic system modelling chemotaxis; in: Self-similar solutions of nonlinear PDE,, Banach Center Publ., 74 (2006), 149. doi: 10.4064/bc74-0-9. Google Scholar

[21]

A. Raczyński, Stability property of the two-dimensional Keller-Segel model,, Asymptot. Anal., 61 (2009), 35. Google Scholar

[22]

T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis,, J. Functional Anal., 191 (2002), 17. doi: 10.1006/jfan.2001.3802. Google Scholar

[23]

J. J. L. Vélazquez, Point Dynamics in a singular limit of the Keller-Segel model 1: Motion of the concentration regions,, SIAM J. Appl. Math., 64 (2004), 1198. doi: 10.1137/S0036139903433888. Google Scholar

show all references

References:
[1]

J. Bedrossian and N. Masmoudi, Existence, uniqueness and Lipschitz dependence for Patlak-Keller-Segel and Navier-Stokes in $\mathbb R^2$ with measure-valued initial data,, Arch. Rational Mech. Anal., 214 (2014), 717. doi: 10.1007/s00205-014-0796-z. Google Scholar

[2]

P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation,, Studia Math., 114 (1995), 181. Google Scholar

[3]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 715. Google Scholar

[4]

P. Biler, Radially symmetric solutions of a chemotaxis model in the plane - the supercritical case, in: Parabolic and Navier-Stokes Equations,, Banach Center Publications, 81 (2008), 31. doi: 10.4064/bc81-0-2. Google Scholar

[5]

P. Biler and L. Brandolese, On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modelling chemotaxis,, Studia Math., 193 (2009), 241. doi: 10.4064/sm193-3-2. Google Scholar

[6]

P. Biler, L. Corrias and J. Dolbeault, Large mass self-similar solutions of the parabolic-parabolic Keller-Segel model,, J. Math. Biology, 63 (2011), 1. doi: 10.1007/s00285-010-0357-5. Google Scholar

[7]

P. Biler, G. Karch, Ph. Laurençot and T. Nadzieja, The $8\pi$-problem for radially symmetric solutions of a chemotaxis model in the plane,, Math. Methods in the Applied Sci., 29 (2006), 1563. doi: 10.1002/mma.743. Google Scholar

[8]

A. Blanchet, E. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model,, J. Functional Anal., 262 (2012), 2142. doi: 10.1016/j.jfa.2011.12.012. Google Scholar

[9]

A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbb R^2$,, Comm. Pure Appl. Math., 61 (2008), 1449. doi: 10.1002/cpa.20225. Google Scholar

[10]

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

[11]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbb R^2$,, Commun. Math. Sci., 6 (2008), 417. Google Scholar

[12]

L. Corrias, M. Escobedo and J. Matos, Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane,, J. Differential Equations, 257 (2014), 1840. doi: 10.1016/j.jde.2014.05.019. Google Scholar

[13]

J. Dolbeault and Ch. Schmeiser, The two-dimensional Keller-Segel model after blow-up,, Discrete Contin. Dyn. Syst., 25 (2009), 109. doi: 10.3934/dcds.2009.25.109. Google Scholar

[14]

Y. Giga, T. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity,, Arch. Rational Mech. Anal., 104 (1988), 223. doi: 10.1007/BF00281355. Google Scholar

[15]

H. Kozono and Y. Sugiyama, Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system,, J. Evol. Equ., 8 (2008), 353. doi: 10.1007/s00028-008-0375-6. Google Scholar

[16]

P.-G. Lemarié-Rieusset, Small data in an optimal Banach space for the parabolic-parabolic and parabolic-elliptic Keller-Segel equations in the whole space,, Adv. Diff. Eq., 18 (2013), 1189. Google Scholar

[17]

S. Luckhaus, Y. Sugiyama and J. J. L. Vélazquez, Measure valued solutions of the 2D Keller-Segel system,, Arch. Rational Mech. Anal., 206 (2012), 31. doi: 10.1007/s00205-012-0549-9. Google Scholar

[18]

N. Mizoguchi, Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane,, Calc. Var., 48 (2013), 491. doi: 10.1007/s00526-012-0558-4. Google Scholar

[19]

N. Mizoguchi and M. Winkler, (2013),, personal communication., (). Google Scholar

[20]

Y. Naito, Asymptotically self-similar solutions for the parabolic system modelling chemotaxis; in: Self-similar solutions of nonlinear PDE,, Banach Center Publ., 74 (2006), 149. doi: 10.4064/bc74-0-9. Google Scholar

[21]

A. Raczyński, Stability property of the two-dimensional Keller-Segel model,, Asymptot. Anal., 61 (2009), 35. Google Scholar

[22]

T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis,, J. Functional Anal., 191 (2002), 17. doi: 10.1006/jfan.2001.3802. Google Scholar

[23]

J. J. L. Vélazquez, Point Dynamics in a singular limit of the Keller-Segel model 1: Motion of the concentration regions,, SIAM J. Appl. Math., 64 (2004), 1198. doi: 10.1137/S0036139903433888. Google Scholar

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