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Finite time blow up and non-uniform bound for solutions to a degenerate drift-diffusion equation with the mass critical exponent under non-weight condition

  • Takayoshi Ogawa
  • Hiroshi WakuiEmail author
Article

Abstract

We consider the non-existence and the non-uniform boundedness of a time global solution to the Cauchy problem of a degenerate drift-diffusion system with the mass critical exponent. If the initial data has negative free energy, then either the corresponding weak solution to the equation does not exist globally in time, or the time global solution does not remain bounded in the energy space. We emphasize that our result does not require any weight assumption on the initial data, and hence, a solution may have an infinite second moment. The proof is based upon the modified virial law and conservation laws and we show that the modified moment functional vanishes for a finite time under the negative energy condition. For a radially symmetric case, the solution blows up in finite time and the mass concentration phenomenon occurs with a sharp lower bound related to the best constant for the Hardy–Littlewood–Sobolev inequality.

Mathematics Subject Classification

Primary 35K65 Secondary 35K45 35B33 35B44 

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Notes

Acknowledgements

The first author is partially supported by JSPS Grant-in-aid for Scientific Research S #25220702 and Challenging Research (Pioneering) #17H06199. The Second author is supported by JSPS Grant-in-aid for Scientific Research S #25220702.

References

  1. 1.
    Bian, S.: A note on the free energy of the Keller–Segel model for subcritical and supercritical cases. Nonlinear Anal. 125, 406–422 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Biler, P.: Existence and nonexistence of solutions for a model of gravitational interaction of particles, III. Colloq. Math. 68, 229–239 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Biler, P.: Local and global solvability of some parabolic systems modeling chemotaxis. Adv. Math. Sci. Appl. 8, 715–743 (1998)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Biler, P., Dolbeault, J.: Long time behavior of solutions to Nernst–Planck and Debye–Hünkel drift-diffusion systems. Ann. Henry Poincaré 1, 461–472 (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Biler, P., Hebisch, W., Nadzieja, T.: The Debye system: existence and large time behavior of solutions. Nonlinear Anal. Theory Methods Appl. 23, 1189–1209 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Biler, P., Nadzieja, T., Stanczy, R.: Nonisothermal systems of self-attracting Fermi–Dirac particles. Banach Cent. Publ. 66, 61–78 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bedrossian, J.: Large mass global solutions for a class of \(L^1\) critical nonlocal aggregation equations and parabolic-elliptic Patlak–Keller–Segel models. Commun. Part. Differ. Equ. 40, 1119–1136 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Blanchet, A., Carrillo, J., Laurençont, P.: Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions. Calc. Val. Part. Differ. Equ. 35, 133–168 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Blanchet, A., Dolbeault, J., Perthame, B.: Two-dimensional Keller–Segel model: optimal critical mass and qualitative properties of the solutions. Electron. J. Differ. Equ. 2006(44), 1–33 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen, L., Liu, J.-G., Wang, J.: Multi-dimensional degenerate Keller–Segel system with critical diffusion exponent \(2n/(n+2)\). SIAM J. Math. Anal. 44, 1077–1102 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cieślak, T., Winkler, M.: Finite-time blow-up in a quasilinear system of chemotaxis. Nonlinearity 21, 1057–1076 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Feireisl, E., Laurençot, P.: Non-isotheral Smoluchowski–Poisson equations as a singular limit of the Navier–Stokes–Fourier–Poisson system. J. Math. Pures Appl. 88, 325–349 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fujita, H.: On the blowing-up of solutions of the Cauchy problem for \(u_t=\Delta u+u^{1+\alpha }\). J. Fac. Sci. Univ. Tokyo Ser I 13, 109–124 (1966)Google Scholar
  14. 14.
    Galaktionov, V.A.: Blow-up for quasilinear heat equations with critical Fujita’s exponents. Proc. R. Soc. Edinb. Sect. A 124, 517–525 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gualdani, M.P., Guillen, N.: Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential. Anal. PDE 9, 1772–1809 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ishige, K., Laurençot, P., Mizoguchi, N.: Blow-up behavior of solutions to a degenerate parabolic–parabolic Keller–Segel system. Math. Ann. 367, 461–499 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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)CrossRefzbMATHGoogle Scholar
  18. 18.
    Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)CrossRefzbMATHGoogle Scholar
  19. 19.
    Kimijima, A., Nakagawa, K., Ogawa, T.: Threshold of global behavior of solutions to a degenerate drift-diffusion system in between two critical exponents. Calc. Var. Part. Differ. Equ. 53, 441–472 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kobayashi, T., Ogawa, T.: Fluid mechanical approximation to the degenerated drift-diffusion system from compressible Navier–Stokes–Poisson system. Indiana Univ. Math. J. 62, 1021–1054 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kurokiba, M., Ogawa, T.: Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type. Differ. Integral Equ. 16, 427–452 (2003)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kim, I., Yao, Y.: The Patlak–Keller–Segel model and its variations: properties of solutions via maximum principle. SIAM J. Math. Anal. 44(2), 568–602 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lieb, E.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. 118, 349–374 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lieb, E., Loss, M.: Analysis, 2nd edn, Amer. Math. Soc. GSM 14 (2001)Google Scholar
  25. 25.
    Laurençot, P., Mizoguchi, N.: Finite time blowup for the parabolic–parabolic Keller–Segel system with critical diffusion. Ann. Inst. Henri Poincaré Anal. 34, 197–220 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Merle, F., Tsutsumi, Y.: \(L^{2}\) concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity. J. Differ. Equ. 84, 205–214 (1990)CrossRefzbMATHGoogle Scholar
  27. 27.
    Mock, M.S.: An initial value problem from semiconductor devise theory. SIAM J. Math. 5(4), 597–612 (1974)CrossRefzbMATHGoogle Scholar
  28. 28.
    Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5, 581–601 (1995)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Nagai, T.: Global existence of solutions to a parabolic system for chemotaxis in two space dimensions. Nonlinear Anal. Theory Methods Appl. 30, 5381–5388 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    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
  31. 31.
    Nagai, T., Ogawa, T.: Global existence of solutions to a parabolic–elliptic system of drift-diffusion type in \(\mathbb{R}^{2}\). Funk. Ekvac. 59, 67–112 (2016)CrossRefzbMATHGoogle Scholar
  32. 32.
    Nagai, T., Senba, T., Suzuki, T.: Chemotactic collapse in a parabolic system of mathematical biology. Hiroshima Math. J. 30, 463–497 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Nagai, T., Senba, T., Yoshida, K.: Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkcial. Ekvac. 40(3), 411–433 (1997)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Ogawa, T.: Decay and asymptotic behavior of a solution of the Keller–Segel system of degenerated and non-degenerated type. Banach Cent. Publ. 74, 161–184 (2006)CrossRefzbMATHGoogle Scholar
  35. 35.
    Ogawa, T.: Asymptotic stability of a decaying solution to the Keller–Segel system of degenerate type. Differ. Integral Equ. 21, 1113–1154 (2008)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Ogawa, T.: The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete Contin. Dyn. Syst. Ser. S 4, 875–886 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Ogawa, T., Tsutsumi, Y.: Blow-up of \(H^1\) solution for the nonlinear Schrödinger equation. J. Differ. Equ. 92, 317–330 (1991)CrossRefzbMATHGoogle Scholar
  38. 38.
    Ogawa, T., Wakui, H.: Non-uniform bound and finite time blow up for solutions to a drift-diffusion equation in higher dimensions. Anal. Appl. 14, 145–183 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Senba, T.: Blowup in infinite time of radial solutions for a parabolic–elliptic system in higher dimensions. Nonlinear Anal. 70, 2549–2562 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Sugiyama, Y.: Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller–Segel system. Differ. Integral Equ. 19, 841–876 (2006)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Sugiyama, Y., Kunii, H.: Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term. J. Differ. Equ. 227, 333–364 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Suzuki, T.: Free Energy and Self-interacting Particles, Progress in Nonlinear Differential Equations and their Applications, vol. 62. Birkhäuser Boston Inc., Boston, MA (2005)Google Scholar
  43. 43.
    Suzuki, T., Takahashi, R.: Degenerate parabolic equations with critical exponent derived from the kinetic theory I, generation of the weak solution. Adv. Differ. Equ. 14, 433–476 (2009)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Suzuki, T., Takahashi, R.: Degenerate parabolic equations with critical exponent derived from the kinetic theory II, blowup threshold. Differ. Integral Equ. 22, 1153–1172 (2009)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Suzuki, T., Takahashi, R.: Degenerate parabolic equation with critical exponent derived from the kinetic theory III, \(\varepsilon \)-regularity. Differ. Integral Equ. 25, 223–250 (2012)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Vázquez, J.L.: The Porous Medium Equation, Mathematical Theory. Oxford Mathematical Monographs. Oxford University Press, Oxford (2006)CrossRefGoogle Scholar
  47. 47.
    Yao, Y.: Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation. Ann. Inst. Henry Poincaré Ann. 31, 81–101 (2014)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Weissler, F.: Local existence and nonexistence for semilinear parabolic equations in \(L^p\). Indiana Univ. Math. J. 29, 79–102 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteTohoku UniversitySendaiJapan

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