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Zeitschrift für angewandte Mathematik und Physik

, Volume 66, Issue 5, pp 2473–2484 | Cite as

Boundedness in a quasilinear fully parabolic Keller–Segel system with logistic source

  • Qingshan ZhangEmail author
  • Yuxiang Li
Article

Abstract

This paper deals with the Neumann boundary value problem for the system
$$\left\{\begin{array}{lll}u_t = \nabla \cdot \left(D(u) \nabla u\right) - \nabla \cdot \left(S(u) \nabla v\right) + f(u), &\quad x \in \Omega, \, t > 0,\\ v_t = \Delta v - v + u, &\quad x \in \Omega, \, t > 0\end{array}\right.$$
in a smooth bounded domain \({\Omega\subset{\mathbb{R}}^n}\) \({(n\geq1)}\), where the functions D(u) and S(u) are supposed to be smooth satisfying \({D(u)\geq Mu^{-\alpha}}\) and \({S(u)\leq Mu^{\beta}}\) with M > 0, \({\alpha\in{\mathbb{R}}}\) and \({\beta\in{\mathbb{R}}}\) for all \({u\geq1}\), and the logistic source f(u) is smooth fulfilling \({f(0)\geq0}\) as well as \({f(u)\leq a-\mu u^{\gamma}}\) with \({a\geq0}\), \({\mu > 0}\) and \({\gamma\geq1}\) for all \({u\geq0}\). It is shown that if
$$\alpha + 2\beta < \left\{\begin{array}{lll}\gamma - 1 + \frac{2}{n}, &\quad {\rm for} \, 1 \leq \gamma < 2,\\ \gamma - 1 + \frac{4}{n + 2}, &\quad {\rm for} \, \gamma \geq 2,\end{array}\right.$$
then for sufficiently smooth initial data, the problem possesses a unique global classical solution which is uniformly bounded.

Keywords

Quasilinear chemotaxis system Logistic source Global solution Boundedness 

Mathematics Subject Classification

35K59 92C17 35K55 

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References

  1. 1.
    Cao X.: Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with logistic source. J. Math. Anal. Appl. 412, 181–188 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cieślak T., Stinner C.: Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions. J. Differ. Equ. 252, 5832–5851 (2012)CrossRefzbMATHGoogle Scholar
  3. 3.
    Cieślak T., Stinner C.: Finite-time blowup in a supercritical quasilinear parabolic–parabolic Keller–Segel system in dimension 2. Acta Appl. Math. 129, 135–146 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cieślak T., Stinner C.: New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models. J. Differ. Equ. 258, 2080–2113 (2015)CrossRefzbMATHGoogle Scholar
  5. 5.
    Hillen T., Painter K.: Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. Appl. Math. 26, 280–301 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Horstmann D., Wang G.: Blow-up in a chemotaxis model without symmetry assumptions. Eur. J. Appl. Math. 12, 159–177 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Horstmann D., Winkler M.: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215, 52–107 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ishida S., Seki K., Yokota T.: Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains. J. Differ. Equ. 256, 2993–3010 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Keller E.F., Segel L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)CrossRefzbMATHGoogle Scholar
  10. 10.
    Lankeit J.: Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source. J. Differ. Equ. 258, 1158–1191 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li X., Xiang Z.: Boundedness in quasilinear Keller–Segel equations with nonlinear sensitivity and logistic source. Discrete Contin. Dyn. Syst. 35, 3503–3531 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mizoguchi N., Souplet P.: Nondegeneracy of blow-up points for the parabolic Keller–Segel system. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 851–875 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nagai T., Senba T., Yoshida K.: Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkcial. Ekvac. 40, 411–433 (1997)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Nakaguchi E., Osaki K.: Global existence of solutions to a parabolic–parabolic system for chemotaxis with weak degradation. Nonlinear Anal. 74, 286–297 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nakaguchi E., Osaki K.: Global solutions and exponential attractors of a parabolic–parabolic system for chemotaxis with subquadratic degradation. Discrete Contin. Dyn. Syst. Ser. B 18, 2627–2646 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Osaki K., Tsujikawa T., Yagi A., Mimura M.: Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal. 51, 119–144 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Osaki K., Yagi A.: Finite dimensional attractor for one-dimensional Keller–Segel equations. Funkcial. Ekvac. 44, 441–469 (2001)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Painter K.J., Hillen T.: Volume-filling and quorum-sensing in models for chemosensitive movement. Can. Appl. Math. Q. 10, 501–543 (2002)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Quittner, P., Souplet, P.: Superlinear parabolic problems. Birkhäuser advanced texts: Basler Lehrbücher. [Birkhäuser advanced texts: Basel textbooks], Birkhäuser Verlag, Basel (2007). Blow-up, global existence and steady statesGoogle Scholar
  20. 20.
    Senba T., Suzuki T.: Parabolic system of chemotaxis: blowup in a finite and the infinite time. Methods Appl. Anal. 8, 349–367 (2001) IMS Workshop on Reaction–Diffusion Systems (Shatin, 1999)Google Scholar
  21. 21.
    Tao Y., Winkler M.: Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity. J. Differ. Equ. 252, 692–715 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang L., Li Y., Mu C.: Boundedness in a parabolic–parabolic quasilinear chemotaxis system with logistic source. Discrete Contin. Dyn. Syst. 34, 789–802 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Winkler M.: Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source. Commun. Partial Differ. Equ. 35, 1516–1537 (2010)CrossRefzbMATHGoogle Scholar
  24. 24.
    Winkler M.: Does a ‘volume-filling effect’ always prevent chemotactic collapse?. Math. Methods Appl. Sci. 33, 12–24 (2010)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Winkler M.: Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction. J. Math. Anal. Appl. 384, 261–272 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Winkler M.: Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system. J. Math. Pures Appl. (9) 100, 748–767 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Winkler M.: Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening. J. Differ. Equ. 257, 1056–1077 (2014)CrossRefzbMATHGoogle Scholar
  28. 28.
    Winkler M., Djie K.C.: Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect. Nonlinear Anal. 72, 1044–1064 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Xiang T.: Boundedness and global existence in the higher-dimensional parabolic–parabolic chemotaxis system with/without growth source. J. Differ. Equ. 258, 4275–4323 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsSoutheast UniversityNanjingPeoples Republic of China

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