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Analysis of nonlinear diffusion equations and related phase transition problems

Research Project

Project/Area Number 12640224
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionWaseda University

Principal Investigator

YAMADA Yoshio  School of Science and Engineering, Professor, 理工学部, 教授 (20111825)

Co-Investigator(Kenkyū-buntansha) HIROSE Munemitsu  Meiji Univ., School of Sci. & Tech., Lecturer, 理工学部, 専任講師 (50287984)
TANAKA Kazunaga  School of Science and Engineering, Professor, 理工学部, 教授 (20188288)
OTANI Mitsuharu  School of Science and Engineering, Professor, 理工学部, 教授 (30119656)
TAKEUCHI Shingo  Gakushuin Univ., Dept. of Math., Assistant Professor, 理学部, 助手 (00333021)
NAKASHIMA Kimie  Tokyo Univ. of Fisheries, Associate Professor, 助教授 (10318800)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥1,300,000 (Direct Cost: ¥1,300,000)
Keywordsreaction-diffusion equation / nonlinear diffusion / global solution / Lotka-Volterra type / positive stationary solution / cross-diffusion / 反応拡散 / 大域的存在 / Shigesadaモデル / 分岐 / 解の多重性
Research Abstract

In our project we have mainly discussed the stationary and non-stationary problems for the following reaction diffusion systems with quasilinear diffusion terms:
(E) u_l = Δ[(1 + αv + γu)u] + uf (u, v), v_l = Δ[(1 + βv + δv)v] + vg (u, v).
This is a well-known system which models the habitat segregation phenomenon between two species. In (E) u, v denote the population densities and f, g represent the interaction between u and v such as Lotka-Volterra competition type or prey-predator type.
(1) Non-stationary problem. When the system has a cross-diffusion effect, the existence result of global solutions was restricted to the two dimensional case. We have proved that, if α, γ > 0 and β = δ = 0, then (E) admits a unique global solution without any restrictions on the space dimension and the amplitude of initial data. Our strategy is to decouple the system and study reaction-diffusion equations separately. We combine parabolic fundamental estimates with energy estimates of solutions of parabolic equation with self-diffusion. This method is also valid for the case δ > 0; so that the global existence is shown when the space dimension is less than six.
(2) Stationary problem. From the view-point of mathematical biology, it is very important to study positive stationary solutions and to know their number. We have tried to get some conditions for the multiplicity of such positive solutions. In particular, the multiple existence is established if interactions are very large in case of competition model with linear diffusion or if one of cross-diffusion is very large in case of prey-predator model.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report

Research Products

(72 results)

All Other

All Publications

  • [Publications] 「研究成果報告書概要(和文)」より

    • Related Report
      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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    • Related Report
      2002 Final Research Report Summary
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    • Related Report
      2002 Final Research Report Summary
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    • Related Report
      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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    • Related Report
      2002 Final Research Report Summary
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    • Related Report
      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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    • Related Report
      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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      2002 Final Research Report Summary
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    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

    • Related Report
      2002 Final Research Report Summary
  • [Publications] 「研究成果報告書概要(欧文)」より

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      2002 Final Research Report Summary
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Published: 2000-04-01   Modified: 2016-04-21  

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