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

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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

Abstract
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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 and 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), 717-801. doi: 10.1007/s00205-014-0796-z.
[2]	P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math., 114 (1995), 181-205.
[3]	P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
[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), 31-42. doi: 10.4064/bc81-0-2.
[5]	P. Biler and L. Brandolese, On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modelling chemotaxis, Studia Math., 193 (2009), 241-261. doi: 10.4064/sm193-3-2.
[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), 1-32. doi: 10.1007/s00285-010-0357-5.
[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), 1563-1583. doi: 10.1002/mma.743.
[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), 2142-2230. doi: 10.1016/j.jfa.2011.12.012.
[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), 1449-1481. doi: 10.1002/cpa.20225.
[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), 32 pp.
[11]	V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbb R^2$, Commun. Math. Sci., 6 (2008), 417-447.
[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), 1840-1878. doi: 10.1016/j.jde.2014.05.019.
[13]	J. Dolbeault and Ch. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Discrete Contin. Dyn. Syst., 25 (2009), 109-121. doi: 10.3934/dcds.2009.25.109.
[14]	Y. Giga, T. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Rational Mech. Anal., 104 (1988), 223-250. doi: 10.1007/BF00281355.
[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), 353-378. doi: 10.1007/s00028-008-0375-6.
[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), 1189-1208.
[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), 31-80. doi: 10.1007/s00205-012-0549-9.
[18]	N. Mizoguchi, Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane, Calc. Var., 48 (2013), 491-505. doi: 10.1007/s00526-012-0558-4.
[19]	N. Mizoguchi and M. Winkler, (2013), personal communication.
[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), 149-160. doi: 10.4064/bc74-0-9.
[21]	A. Raczyński, Stability property of the two-dimensional Keller-Segel model, Asymptot. Anal., 61 (2009), 35-59.
[22]	T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis, J. Functional Anal., 191 (2002), 17-51. doi: 10.1006/jfan.2001.3802.
[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), 1198-1223. doi: 10.1137/S0036139903433888.

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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), 717-801. doi: 10.1007/s00205-014-0796-z.
[2]	P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math., 114 (1995), 181-205.
[3]	P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
[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), 31-42. doi: 10.4064/bc81-0-2.
[5]	P. Biler and L. Brandolese, On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modelling chemotaxis, Studia Math., 193 (2009), 241-261. doi: 10.4064/sm193-3-2.
[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), 1-32. doi: 10.1007/s00285-010-0357-5.
[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), 1563-1583. doi: 10.1002/mma.743.
[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), 2142-2230. doi: 10.1016/j.jfa.2011.12.012.
[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), 1449-1481. doi: 10.1002/cpa.20225.
[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), 32 pp.
[11]	V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbb R^2$, Commun. Math. Sci., 6 (2008), 417-447.
[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), 1840-1878. doi: 10.1016/j.jde.2014.05.019.
[13]	J. Dolbeault and Ch. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Discrete Contin. Dyn. Syst., 25 (2009), 109-121. doi: 10.3934/dcds.2009.25.109.
[14]	Y. Giga, T. Miyakawa and H. Osada, Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. Rational Mech. Anal., 104 (1988), 223-250. doi: 10.1007/BF00281355.
[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), 353-378. doi: 10.1007/s00028-008-0375-6.
[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), 1189-1208.
[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), 31-80. doi: 10.1007/s00205-012-0549-9.
[18]	N. Mizoguchi, Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane, Calc. Var., 48 (2013), 491-505. doi: 10.1007/s00526-012-0558-4.
[19]	N. Mizoguchi and M. Winkler, (2013), personal communication.
[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), 149-160. doi: 10.4064/bc74-0-9.
[21]	A. Raczyński, Stability property of the two-dimensional Keller-Segel model, Asymptot. Anal., 61 (2009), 35-59.
[22]	T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis, J. Functional Anal., 191 (2002), 17-51. doi: 10.1006/jfan.2001.3802.
[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), 1198-1223. doi: 10.1137/S0036139903433888.

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