Advertisement

Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation

  • Michael WinklerEmail author
Article

Abstract

We consider radially symmetric solutions of the Keller–Segel system with generalized logistic source given by
$$\begin{aligned} \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) + \lambda u - \mu u^\kappa , \\ 0 = \Delta v - v + u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$
under homogeneous Neumann boundary conditions in the ball \(\Omega =B_R(0) \subset \mathbb {R}^n\) for \(n\ge 3\) and \(R>0\), where \(\lambda \in \mathbb {R}, \mu >0\) and \(\kappa >1\). Under the assumption that
$$\begin{aligned} \kappa < \left\{ \begin{array}{ll} \frac{7}{6} &{}\quad \text {if } n\in \{3,4\}, \\ 1+ \frac{1}{2(n-1)} &{}\quad \text {if } n \ge 5, \end{array} \right. \end{aligned}$$
a condition on the initial data is derived which is seen to be sufficient to ensure the occurrence of finite-time blow-up for the corresponding solution of (\(\star \)). Moreover, this criterion is shown to be mild enough so as to allow for the conclusion that in fact any positive continuous radial function on \(\overline{\Omega }\) is the limit in \(L^1(\Omega )\) of a sequence \((u_{0k})_{k\in \mathbb {N}}\) of continuous radial initial data which are such that for each \(k\in \mathbb {N}\) the associated initial-boundary value problem for (\(\star \)) exhibits a finite-time explosion phenomenon in the above sense. In particular, this apparently provides the first rigorous detection of blow-up in a superlinearly dampened but otherwise essentially original Keller–Segel system in the physically relevant three-dimensional case.

Keywords

Chemotaxis Logistic source Finite-time blow-up 

Mathematics Subject Classification

35B44 (primary) 35K65 92C17 (secondary) 

Notes

Acknowledgements

The author acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Analysis of chemotactic cross-diffusion in complex frameworks.

References

  1. 1.
    Bai, X., Winkler, M.: Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics. Indiana Univ. Math. J. 65, 553–583 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bianchi, A., Painter, K.J., Sherratt, J.A.: Spatio-temporal models of lymphangiogenesis in wound healing. Bull. Math. Biol. 78, 1904–1941 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Black, T., Lankeit, J., Mizukami, M.: On the weakly competitive case in a two-species chemotaxis model. IMA J. Appl. Math. 81, 860–876 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cao, X.: Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with logistic source. J. Math. Anal. Appl. 412, 181–188 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cao, X.: Boundedness in a three-dimensional chemotaxis–haptotaxis model. Z. Angew. Math. Phys. 67, 11 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cao, X., Zheng, S.: Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source. Math. Methods Appl. Sci. 37, 2326–2330 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chaplain, M.A.J., Lolas, G.: Mathematical modelling of cancer invasion of tissue: the role of the urokinase plasminogen activation system. Math. Models Methods Appl. Sci. 15, 1685–1734 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cieślak, T., Winkler, M.: Finite-time blow-up in a quasilinear system of chemotaxis. Nonlinearity 21, 1057–1076 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Djie, K., Winkler, M.: Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect. Nonlinear Anal. TMA 72(2), 1044–1064 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    He, X., Zheng, S.: Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source. J. Math. Anal. Appl. 436, 970–982 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Herrero, M.A., Velázquez, J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Scu. Norm. Super. Pisa Cl. Sci. 24, 663–683 (1997)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58(1), 183–217 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jäger, W., Luckhaus, S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329, 819–824 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kang, K., Stevens, A.: Blowup and global solutions in a chemotaxis-growth system. Nonlinear Anal. TMA 135, 57–72 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kiselev, A., Ryzhik, L.: Biomixing by chemotaxis and enhancement of biological reactions. Comm. Partial Differ. Equ. 37(1–3), 298–318 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lankeit, J.: Chemotaxis can prevent thresholds on population density. Discrete Contin. Dyn. Syst. B 20, 1499–1527 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lankeit, J.: Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source. J. Differ. Equ. 258, 1158–1191 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lankeit, J., Mizukami, M.: How far does small chemotactic interaction perturb the Fisher–KPP dynamics? J. Math. Anal. Appl. 452, 429–442 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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
  20. 20.
    Meral, G., Stinner, C., Surulescu, C.: On a multiscale model involving cell contractivity and its effects on tumor invasion. Discrete Contin. Dyn. Syst. Ser. B 20, 189–213 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mimura, M., Tsujikawa, T.: Aggregating pattern dynamics in a chemotaxis model including growth. Physica A 230, 499–543 (1996)CrossRefGoogle Scholar
  22. 22.
    Nadin, G., Perthame, B., Ryzhik, L.: Traveling waves for the Keller–Segel system with Fisher birth terms. Interfaces Free Bound. 10, 517–538 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5, 581–601 (1995)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Nagai, T.: Blowup of nonradial solutions to parabolic–elliptic systems modeling chemotaxis in two-dimensional domains. J. Inequal. Appl. 6, 37–55 (2001)MathSciNetzbMATHGoogle Scholar
  25. 25.
    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
  26. 26.
    Painter, K.J., Hillen, T.: Spatio-temporal chaos in a chemotaxis model. Physica D 240, 363–375 (2011)CrossRefzbMATHGoogle Scholar
  27. 27.
    Painter, K.J., Maini, P.K., Othmer, H.G.: Complex spatial patterns in a hybrid chemotaxis reaction–diffusion model. J. Math. Biol. 41(4), 285–314 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Salako, R., Shen, W.: Global existence and asymptotic behavior of classical solutions to a parabolic–elliptic chemotaxis system with logistic source on \(\mathbb{R}^N\). J. Differ. Equ. 262, 5635–5690 (2017)CrossRefzbMATHGoogle Scholar
  29. 29.
    Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theor. Biol. 79, 83–99 (1979)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Strohm, S., Tyson, R.C., Powell, J.A.: Pattern formation in a model for mountain pine beetle dispersal: linking model predictions to data. Bull. Math. Biol. 75, 1778–1797 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tao, Y., Winkler, M.: A chemotaxis–haptotaxis model: the roles of porous medium diffusion and logistic source. SIAM J. Math. Anal. 43, 685–704 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Tao, Y., Winkler, M.: Boundedness and stabilization in a multi-dimensional chemotaxis–haptotaxis model. Proc. R. Soc. Edinb. Sect. A 144, 1067–1084 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Tao, Y., Winkler, M.: Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system. Z. Angew. Math. Phys. 66, 2555–3573 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Tello, J.I., Winkler, M.: A chemotaxis system with logistic source. Comm. Partial Differ. Equ. 32(6), 849–877 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Viglialoro, G.: Very weak global solutions to a parabolic–parabolic chemotaxis-system with logistic source. J. Math. Anal. Appl. 439, 197–212 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Viglialoro, G.: Boundedness properties of very weak solutions to a fully parabolic chemotaxis system with logistic source. Nonlinear Anal. Real World Appl. 34, 520–535 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Viglialoro, G., Woolley, T.E.: Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete Contin. Dyn. Syst. B (2017).  https://doi.org/10.3934/dcdsb.201-7199 Google Scholar
  38. 38.
    Wang, L., Li, Y., Mu, C.: Boundedness in a parabolic–parabolic quasilinear chemotaxis system with logistic source. Discrete Contin. Dyn. Syst. A 34, 789–802 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wang, L., Mu, C., Hu, X., Tian, Y.: Boundedness in a quasilinear chemotaxis–haptotaxis system with logistic source. Math. Methods Appl. Sci. 40, 3000–3016 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Wang, L., Mu, C., Zheng, P.: On a quasilinear parabolic–elliptic chemotaxis system with logistic source. J. Differ. Equ. 256, 1847–1872 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Winkler, M.: Chemotaxis with logistic source: very weak global solutions and boundedness properties. J. Math. Anal. Appl. 348, 708–729 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Winkler, M.: Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source. Comm. Partial Differ. Equ. 35, 1516–1537 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Winkler, M.: Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction. J. Math. Anal. Appl. 384, 261–272 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Winkler, M.: Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system. J. Math. Pures Appl. 100, 748–767 (2013). arXiv:1112.4156v1
  45. 45.
    Winkler, M.: How far can chemotactic cross-diffusion enforce exceeding carrying capacities? J. Nonlinear Sci. 24, 809–855 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Winkler, M.: Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems. Discrete Contin. Dyn. Syst. Ser. B 22, 2777–2793 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Winkler, M.: Blow-up profiles and life beyond blow-up in the fully parabolic Keller–Segel system (preprint)Google Scholar
  49. 49.
    Woodward, D.E., Tyson, R., Myerscough, M.R., Murray, J.D., Budrene, E.O., Berg, H.C.: Spatiotemporal patterns generated by Salmonella typhimurium. Biophys. J. 68(5), 2181–2189 (1995)CrossRefGoogle Scholar
  50. 50.
    Yang, C., Cao, X., Jiang, Z., Zheng, S.: Boundedness in a quasilinear fully parabolic Keller–Segel system of higher dimension with logistic source. J. Math. Anal. Appl. 430, 585–591 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Zhang, Q., Li, Y.: Global boundedness of solutions to a two-species chemotaxis system. Z. Angew. Math. Phys. 66, 83–93 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Zhao, X., Zheng, S.: Global boundedness to a chemotaxis system with singular sensitivity and logistic source. Z. Angew. Math. Phys. 68, 2 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Zheng, P., Mu, C., Hu, X.: Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete Contin. Dyn. Syst. A 35, 2299–2323 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Zheng, J.: Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source. J. Differ. Equ. 259, 120–140 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Zheng, J.: Boundedness and global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with nonlinear logistic source. J. Math. Anal. Appl. 450, 1047–1061 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Zheng, J.: A note on boundedness of solutions to a higher-dimensional quasi-linear chemotaxis system with logistic source. Z. Angew. Math. Mech. 97, 414–421 (2017)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Zheng, J., Wang, Y.: Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source. Comput. Math. Appl. 72, 2604–2619 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PaderbornPaderbornGermany

Personalised recommendations