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Structure of solutions to the system of equations describing chemotactic aggregation of cellular slime molds

Research Project

Project/Area Number 13640216
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionHIROSHIMA UNIVERSIlY

Principal Investigator

YOSHIDA Kiyoshi  Hiroshima Univ., Faculty of Integrated Arts and Sciences, Prof., 総合科学部, 教授 (80033893)

Co-Investigator(Kenkyū-buntansha) USAMI Hiroyuki  Hiroshima Univ., Faculty of Integrated Arts and Sciences, Ass. Prof., 総合科学部, 助教授 (90192509)
SHIBATA Tetsutaro  Hiroshima Univ., Faculty of Integrated Arts and Sciences, Ass. Prof., 総合科学部, 助教授 (90216010)
NAGAI Toshitaka  Hiroshima Univ., Graduate School of Science, Prof., 大学院・理学研究科, 教授 (40112172)
NAITO Yuki  Kobe Univ., Faculty of Engineering, 工学部, 助教授 (10231458)
水田 義弘  広島大学, 総合科学部, 教授 (00093815)
田中 和永  早稲田大学, 理工学部, 教授 (20188288)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2003: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2001: ¥1,200,000 (Direct Cost: ¥1,200,000)
KeywordsKeller-Segel system / Chirdress-Percus conjecture / global blanch / locations of blow up points / self-similar solution / elliptic equations / Lieuville type theorem / radially symmetrical solution / 粘菌形成方程式系 / 楕円型方程式 / 固有値問題 / 大域分岐 / Harnack型不等式 / 大腸菌コロニー / 半群理論 / 楕円形方程式 / 解の爆発 / Liouville型定理 / 球対称性
Research Abstract

Keller and Segel derived the mathematical model describing chemotactic aggregation of cellular slime molds which move toward high cocentrations of chemical substance. So this model is called as the Keller-Segel system. Chirdress and Percus conjectured that there exists a threshold number (8π) such that if an initial value is smaller than 8π, the solutions exist globally in time, on the other hand if an initial value is greater than 8π, the chemotactic collapse can occure. Our objective is to solve this conjecture. We study the Keller-Segel system in R^2 because this system is rarely studied in hole space. Especially we treated the self-similar solution and solved almost positively the Chirdress and Percus conjectuer. For this purpose we encountered several problems and solved these. We enumerate these problems : (1)determine decay order of the self-similar solutions (2)derive the Lieuville type theorem (3)reduction to one elliptic equation from an system of two elliptic equations (4)show the radial symmetry of solutions by using the moving plane method (5)determine the global blanch of solutions (6)solve the Chirdress and Percus conjectuer.

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (40 results)

All Other

All Publications (40 results)

  • [Publications] K.Yoshida: "Self-similar solutions of chemotaxic system, Journal of Nonlinear Analysis"Journal of Nonlinear Analysis. 47. 813-824 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Naito, T.Suzuki, K.Yoshida: "Self-similar solutions to a parabolic system modeling chemotaxis"Journal of Differential Equations. 184. 386-421 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nagai: "Global existence and blowup of solutions to a chemotaxis system"Ninlinear Analysis. 47. 777-787 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nagai, R.Syukuinn, M.Umesako: "Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in R^n"Funkcialij Ekvacioj. 46. 383-407 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Shibata: "Three-term special asymptotics for nonlinear Strum-Lieuville problems"Nonlinear Differential Equations Appl. 9. 75-90 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Teramoto, H.Usami: "A Lieuville type theorem for semilinear elliptic systems"Pacific J.Math.. 204(1). 247-255 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Naito, H.Usami: "Oscillation criteria for quasilinear elliptic equations"Nonlinear Analysis. 46. 629-652 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Yoshida: "Self-similar solutions of chemotaxic system"Journal of Nonlinear Analysis. 47. 813-824 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Naito, T.Suzuki, K.Yoshida: "Self-similar solutions to a parabolic system modeling chemotaxis"J. Duff Equations. 184. 386-342 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nagai: "Global existence and blowup of solutions to a chemotaxis system"Nonlinear Analysis. 47. 777-787 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] G.Harada, T.Nagai, T.Senba, T.Suzuki: "Concentration lemma, Brezis-Merle type inequality, and a parabolic system of chemotaxis"Adv. Differential Equations. 6. 1255-1280 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nagai, R.Syukuinn, M.Umesako: "Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in R^n"Funkcialij Ekvacioj. 46. 383-407 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Shibata: "Three-term special asymptotics for nonlinear Strum-Lieuville problems"Nonlinear Duff. Equations Appl. 9. 75-90 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Teramoto, H.Usami: "A Lieuville type theorem for semilinear elliptic systems"Pacific J. Math.. 204. 247-255 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Naito, H.Usami: "Oscillation criteria for quasilinear elliptic equations"Nonlinear Analysis. 46. 629-652 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Futamura, Y.Mizuta: "Lindelof theorems for monotone Sobolev functions"Ann. Acad. Sci. Fenn. Math.. 28. 271-277 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y Mizuta, T.Shimomura: "On semi-fine limits at infinity for Riesz potentials and monotone BLD functions"Potential Analysis.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Yoshida: "Self-similar solutions of chemotaxic system, Journal of Nonlinear Analysis"Journal of Nonlinear Analysis. 47. 813-824 (2001)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Naito, T.Susuki, K.Yoshida: "Self-similar solutions to a parabolic system modeling chemotaxis"Journal of Differential Equations. 184. 386-421 (2001)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Nagai: "Global existence and blowup of solutions to a chemotaxis system"Ninlinear Analysis. 47. 777-787 (2001)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Nagai, R.Syukuinn, M.Umesako: "Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in R^n"Funkcialij Ekvacioj. 46. 383-407 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Shibata: "Three-term special asymptotics for nonlinear Strum-Lieuville problems"Nonlinear Differential Equations Appl. 9. 75-90 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Shibata: "Precise spectral asymptotics for nonlinear Strum-Lieuville problems quations"J.Differental Equations. 180(2). 374-394 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Teramoto, H.Usami: "A Lieuville type theorem for semilinear elliptic systems"Pacific J.Math.. 204(1). 247-255 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Naito, H.Usami: "Oscillation criteria for quasilinear elliptic equations"Nonlinear Analysis. 46(5). 629-652 (2001)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Yoshida: "Self-similar solutions of chemotaxic system, Journal of Nonlinear Analysis"Journal of Nonlinear Analysis. 47. 813-824 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Naito, T.Suzuki, K.Yoshida: "Self-similar solutions to a parabolic system modeling chemotaxis"Journal of Differential Equations. 184. 386-421 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Mizuta, T.shimomura: "Continuity and differentiablity for weighted Sobolev spaces"Proc Amer. Math. Soc.. 130. 2985-2994 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Futamura, Y.Mizuta: "Existence of functions in weighted Sobolev spaces"Nagoya Math. J.. 164. 75-88 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Shibata: "Three-term special asymptotics for nonlinear Strum-Lieuville problems"Nonlinear Differential Equations Appl. 9. 75-90 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Shibata: "Precise spectral asymptotics for nonlinear Sturm-Lieuville problems equations"J. Differental Equations. 180(2). 374-394 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Teramoto, H.Usami: "A Lieuville type theorem for semilinear elliptic systems"Pacific J. Math.. 204(1). 247-255 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Naito, H.Usami: "Oscillation criteria for quasilinear elliptic equations"Nonlinear Analysis. 46(5). 629-652 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Muramoto, Y.Naito, K.Yoshida: "Existence of self-similar solutions to a parabolic system modelling chemotaxis"Japan Journal of Industrial and Applied Mathematics. 17. 427-751 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Yoshida: "Self-similar solutions of chemotaxic system, Journal of Nonlinear Analysis"Journal of Nonlinear Analysis. 47. 813-824 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Naagai: "Behavior of solutions to a parabolic-elliptic system modelling chemotaxis"J. Korean Math. Soc.. 37. 721-733 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Futamura, K.Kishi, Y.Mizuta: "A generalization of the Liouville theorem to polyharmonic functions"J. Math. Soc. Japan. 53. 113-118 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Shibata: "Precise asymptotics of nonlinear elliptic two-parameter problems"Nonlinear Analysis. 43(1). 75-90 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Naito, H.Usami: "Oscillation criteria for quasilinear elliptic equations"Nonlinear Analysis. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Adachi, T.Adachi: "Multiple positive solutions for nonhomogeneous elliptic equations"Nonlinear Analysis. 47(6). 3783-3793 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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