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ANALYSIS OF SYSTEMS OF DIFFERENTIAL EQUATIONS HAVING SEVERAL NONLOCAL TERMS AND APPLICATIONS

Research Project

Project/Area Number 15540172
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionSHIMANE UNIVERSITY

Principal Investigator

KATO Nobuyuki  Shimane University, Dept.Math, Associate Professor, 総合理工学部, 助教授 (40177423)

Co-Investigator(Kenkyū-buntansha) FURUMOCHI Tetsuo  Shimane University, Dept.Math, Professor, 総合理工学部, 教授 (40039128)
AIKAWA Hiroaki  Shimane University, Dept.Math, Professor, 総合理工学部, 教授 (20137889)
SUGIE Jitsuro  Shimane University, Dept.Math, Professor, 総合理工学部, 教授 (40196720)
MACHIHARA Shuji  Shimane University, Dept.Math, Research Associate, 総合理工学部, 助手 (20346373)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2003: ¥1,500,000 (Direct Cost: ¥1,500,000)
Keywordsnonlocal / population dynamics / size structured / harvesting / time dependent / transport equation / muscle contraction / 非局所非線形成長率
Research Abstract

First, we have investigated size structured population models where the growth rate, the mortality rate and the fertility rate depend on size and total population weighted differently each other. We obtained the existence, the representation, and the uniqueness of solutions. By using Schauder's fixed point theorem, we obtained the existence of solutions in a general model. For the uniqueness, we imposed some stronger conditions on the mortality rate and the fertility rate. Unfortunately, it is not known whether the stronger conditions can be removed or not. The results are published in the journal, J.Math.Anal.Appl.
Next, we have investigated a general time-dependent model of size structured population dynamics, where the growth rate depends on the size and time. We obtained the existence, positivity, and the uniqueness of solutions. The study of the time dependent model is needed to consider nonlinear harvesting problems.
In fact, we applied the results to a nonlinear harvesting problem of size structured population models. Our model has the harvesting rate depending only on time, but the growth rate depends on the size and time and the fertility rate depends on the weighted total population.
We gave a presentation about this result at the international conference held at Florida in 2004.
Further, we have developed the theory of nonlinear transport equations with nonlocal terms prescribed by a multi-valued function and applied the results to nonlinear four state models of muscle contraction.
Here, we obtained not only the existence and uniqueness results, but also the results of the support and the boundedness of the solution. The results will be published in the journal, Adv.Math.Sci.Anal.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (8 results)

All 2004 Other

All Journal Article (4 results) Publications (4 results)

  • [Journal Article] A general model of size-dependent population dynamics with nonlinear growth rate2004

    • Author(s)
      N.Kato
    • Journal Title

      J.Math.Anal.Appl. 297

      Pages: 234-256

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] A system of nonlinear nonlocal transport equations related to muscle contraction mechanism

    • Author(s)
      N.Kato
    • Journal Title

      Adv.Math.Sci.Appl. (発表予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A system of nonlinear nonlocal transport equations related to muscle contraction mechanism

    • Author(s)
      N.Kato
    • Journal Title

      Adv.Math.Sci.Appl. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A system of nonlinear nonlocal transport equations related to muscle contraction mechanism

    • Author(s)
      N.Kato
    • Journal Title

      Adv.Math.Sci.Appl. (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Publications] T.Furumochi, S.Murakami, Y.Nagabuchi: "Stabilities in Volterra difference equations on a Banach space"Fields Institute Communications. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Aikawa: "Positive harmonic functions of finite order in a Denjoy type domain"Proc.Amer.Math.Soc.. 131. 3873-3881 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] J.Sugie, N.Yamaoka: "Oscillation of solutions of second-order nonlinear self-adjoint differential equations"J.Math.Anal.Appl.. 291. 99-116 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Machihara, K.Nakanishi, T.Ozawa: "Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation"Revista Matematica Iberoamericana. 19. 179-194 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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