Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane

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Abstract

This paper is concerned with the Cauchy problem for the Keller–Segel system
$$\left\{\begin{array}{l@{\quad}l}u_t = \nabla \cdot (\nabla u - u \nabla v) & \hbox{in } {\bf R}^{2} \times(0,\infty),\\v_t = \Delta v - \lambda v + u & \hbox{ in } {\bf R}^2 \times(0,\infty),\\u(x,0) = u_0 (x) \geq 0, \; v(x,0) = v_0 (x) \geq 0 & \hbox{ in} {\bf R}^2\end{array}\right.$$
with a constant λ ≥ 0, where \({(u_0, v_0) \in (L^1 ({\bf R}^2) \cap L^\infty ({\bf R}^2) ) \times (L^1 ({\bf R}^2) \cap H^1 ({\bf R}^2))}\). Let
$$m (u_0;{\bf R}^2) = \int\limits_{{\bf R}^2} u_0 (x) dx$$
. The same method as in [9] yields the existence of a blowup solution with m (u 0; R 2) > 8π. On the other hand, it was recently shown in [7] that under additional hypotheses \({u_0 \log (1 + |x|^2) \in L^1 ({\bf R}^2)}\) and \({u_0 \log u_0 \in L^1 ({\bf R}^2)}\), any solution with m(u 0; R 2) < 8π exists globally in time. In[18], the extra assumptions were taken off, but the condition on mass was restricted to m (u 0; R 2) < 4π. In this paper, we prove that any solution with m (u 0; R 2) < 8π exists globally in time under no extra conditions. Furthermore the global existence of solutions is obtained under some condition on u 0 also in the critical case m (u 0; R 2) = 8π.

Mathematics Subject Classification (2010)

35B44 35K45 92C17 
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References

  1. 1.
    Biler P.: Local and global solvability of some parabolic systems modeling chemotaxis. Adv. Math. Sci. Appl. 8, 715–743 (1998)MathSciNetMATHGoogle Scholar
  2. 2.
    Biler P., Nadzieja T.: Existence and nonexistence of solutions for a model of gravitational interactions of particles. Colloquium Math. 66, 319–334 (1994)MathSciNetMATHGoogle Scholar
  3. 3.
    Biler P., Karch G., Laurençot P., Nadzieja T.: The 8π problem for radially symmetric solutions of a chemotaxis model in a disk. Nonlinear Anal. 27, 133–147 (2006)MATHGoogle Scholar
  4. 4.
    Biler P., Karch G., Laurençot P., Nadzieja T.: The 8π problem for radially symmetric solutions of a chemotaxis model in the plane. Math. Methods Appl. Sci. 29, 1563–1583 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Blanchet A., Carrillo J.A., Masmoudi N.: Infinite time aggregation for the critical Patlak-Keller–Segel model in R 2. Commun. Pure Appl. Math. 61, 1449–1481 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Biler P., Corrias L., Dolbeault J.: Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis. J. Math. Biol. 63, 1–32 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Calvez V., Corrias L.: The parabolic–parabolic Keller–Segel model in R 2. Commun. Math. Sci. 6, 417–447 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gajewski H., Zacharias K.: Global behavior of a reaction-diffusion system modeling chemotaxis. Math. Nachr. 195, 77–114 (1998)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Herrero M.A., Velázquez J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Scoula Norm. Sup. Pisa IV 35, 633–683 (1997)Google Scholar
  10. 10.
    Horstmann D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences I. Jahresber. Deutsch. Math. -Verein. 105, 103–165 (2003)MathSciNetMATHGoogle Scholar
  11. 11.
    Jäger W., Luckhaus S.: On explosions of solutions to a system of partial differential equations modeling chemotaxis. Trans. Am. Math. Soc. 329, 819–824 (1992)MATHGoogle Scholar
  12. 12.
    Keller E.F., Segel L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)CrossRefMATHGoogle Scholar
  13. 13.
    Mizoguchi, N., Velázquez, J.J.L.: Forward self-similar solution to the Keller–Segel system with mass greater than 8π (in preparation)Google Scholar
  14. 14.
    Moser J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1971)CrossRefGoogle Scholar
  15. 15.
    Nagai T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5, 581–601 (1995)MathSciNetMATHGoogle Scholar
  16. 16.
    Nagai T.: Blowup of nonradial solutions to parabolic–elliptic systems modeling chemotaxis in two dimensional domains. J. Inequal. Appl. 6, 37–55 (2001)MathSciNetMATHGoogle Scholar
  17. 17.
    Nagai T.: Global existence and decay estimates of solutions to a parabolic–elliptic system of drift-diffusion type in R 2. Differ. Integral Equ. 24, 29–68 (2010)MathSciNetGoogle Scholar
  18. 18.
    Nagai T., Ogawa T.: Brezis–Merle inequalities and application to the global existence of the Cauchy problem of the Keller–Segel system. Commun. Contemp. Math. 13, 795–812 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Nagai T., Senba T., Yoshida K.: Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkc. Ekvac. 40, 411–433 (1997)MathSciNetMATHGoogle Scholar
  20. 20.
    Nagai T., Senba T., Suzuki T.: Chemotactic collapse in parabolic system of mathematical biology. Hiroshima Math. J. 30, 463–497 (2000)MathSciNetMATHGoogle Scholar
  21. 21.
    Ohtsuka H., Senba T., Suzuki T.: Blowup in infinite time in the simplified system of chemotaxis. Adv. Math. Sci. Appl. 17, 445–472 (2007)MathSciNetMATHGoogle Scholar
  22. 22.
    Trudinger N.S.: On imbedding into Orlicz spaces and some applications. J. Math. Mech. 17, 473–484 (1967)MathSciNetMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsTokyo Gakugei UniversityKoganeiJapan
  2. 2.Precursory Research for Embryonic Science and Technology (PRESTO)Japan Science and Technology Agency (JST)KawaguchiJapan

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