Skip to main content
SpringerLink
Log in

A free boundary problem in ecology

  • Published:
Japan Journal of Applied Mathematics Aims and scope Submit manuscript

Abstract

This article is concerned with a free boundary problem for semilinear parabolic equations, which describes the habitat segregation phenomenon in population ecology. The main purpose is to show the global existence, uniqueness, regularity and asymptotic behavior of solutions for the problem. The asymptotic stability or instability of each solution is completely determined using the comparison theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. D. Aronson, M. G. Crandall and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal.,6 (1982), 1001–1022.

    Article  MATH  MathSciNet  Google Scholar 

  2. H. Brézis, Opérateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert. Math. Studies, 5, North-Holland, Amsterdam, 1973.

    Google Scholar 

  3. J. R. Cannon and M. Primicerio, A two phase Stefan problem with flux boundary conditions. Ann. Mat. Pura Appl.,88 (1971), 193–205.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. R. Cannon, D. B. Henry and D. B. Kotlow, Classical solutions of the one-dimensional two-phase Stefan problem. Ann. Mat. Pura Appl.,107 (1975), 311–341.

    Article  MathSciNet  Google Scholar 

  5. J. R. Cannon and E. DiBenedetto, On the existence of weak-solutions to ann-dimensional Stefan problem with nonlinear boundary conditions. SIAM J. Math. Anal.,11 (1980), 632–645.

    Article  MATH  MathSciNet  Google Scholar 

  6. L. C. Evans, A free boundary problem: The flow of two immiscible fluids in a one-dimensional porous medium, I. Indiana Univ. Math. J.,26 (1977), 915–932.

    Article  MATH  MathSciNet  Google Scholar 

  7. L. C. Evans, A free boundary problem: The flow of two immiscible fluids in a one-dimensional porous medium, II. Indiana Univ. Math. J.,27 (1978), 93–111.

    Article  MATH  MathSciNet  Google Scholar 

  8. L. C. Evans and D. B. Kotlow, One dimensional Stefan problems with quasilinear heat conduction. Preprint.

  9. A. Friedman, Remarks on the maximum priciple for parabolic equations and its applications. Pacific J. Math.,8 (1958), 201–211.

    MATH  MathSciNet  Google Scholar 

  10. A. Friedman, The Stefan problem in several space variables. Trans. Amer. Math. Soc.,133 (1968), 51–87: Correction. Ibid.,142 (1969), 557.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Friedman, Partial Differential Equations. Robert E. Krieger Publ. Co., New York, 1976.

    Google Scholar 

  12. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. Vol. 840, Springer-Verlag, Berlin, 1981.

    Google Scholar 

  13. H. Ishii, On a certain estimate of the free boundary in the Stefan problem. J. Differential Equations,42 (1981), 106–115.

    Article  MATH  MathSciNet  Google Scholar 

  14. S. L. Kamenomostskaja, On Stefan’s problem. Mat. Sb.,53 (1961), 489–514.

    MathSciNet  Google Scholar 

  15. T. Kato, Abstract evolution equations of parabolic type in Banach and Hilbert spaces. Nagoya Math. J.,19 (1961), 93–125.

    MATH  MathSciNet  Google Scholar 

  16. N. Kenmochi, Free boundary problems of the Stefan-type for nonlinear parabolic equations with obstacles. Boll. Un. Mat. Ital. B, (6)2 (1983), 171–191.

    MathSciNet  Google Scholar 

  17. O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type. Transl. Math. Monogr.,23, Amer. Math. Soc., Providence, R.I., 1968.

  18. H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations. J. Math. Kyoto Univ.,18 (1978), 221–227.

    MATH  MathSciNet  Google Scholar 

  19. H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. RIMS Kyoto Univ.,15 (1979), 401–454.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Mimura, Y. Yamada and S. Yotsutani, In preparation.

  21. T. Nagai and M. Mimura, Some nonlinear degenerate diffusion equations related to population dynamics. J. Math. Soc. Japan,35 (1983), 539–562.

    Article  MATH  MathSciNet  Google Scholar 

  22. L. Nirenberg, A strong maximum principle for parabolic equations. Comm. Pure Appl. Math.,6 (1953), 167–177.

    Article  MATH  MathSciNet  Google Scholar 

  23. N. Niezgodka, I. Pawlow and A. Visintin, Remarks on the paper by A. Visintin “Sur le problème de Stefan avec flux non linéaire”. Boll. Un. Mat. Ital. C, (5)18 (1981), 87–88.

    MathSciNet  Google Scholar 

  24. A. Okubo, Diffusion and Ecological Problems: Mathematical Models. Springer-Verlag, Berlin, 1980.

    MATH  Google Scholar 

  25. L. I. Rubinstein, The Stefan Problem. Transl. Math. Monogr. 27, Amer. Math. Soc., Providence, R. I., 1971.

  26. D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ.. Math. J.,21 (1972), 979–1000.

    MATH  MathSciNet  Google Scholar 

  27. D. G. Schaeffer, A new proof of the infinite differentiability of the free boundary in the Stefan problem. J. Differential Equations,20 (1976), 266–269.

    Article  MATH  MathSciNet  Google Scholar 

  28. I. Stakgold and L. E. Payne, Nonlinear problems in nuclear reactor analysis. Nonlinear Problems in Physical Science and Biology. Lecture Notes in Math. 322, Springer-Verlag, Berlin, 1973.

    Book  Google Scholar 

  29. H. Tanabe, Equations of Evolution. Pitman, London, 1979.

    MATH  Google Scholar 

  30. Y. Yamada, On evolution equations generated by subdifferential operators. J. Fac. Sci. Univ. Tokyo, Sect. IA, Math.,23 (1976), 491–515.

    MATH  MathSciNet  Google Scholar 

  31. M. Yamaguti and T. Nogi, The Stefan Problem. Sangyo-Tosho, Tokyo, 1977 (in Japanese).

    Google Scholar 

  32. S. Yotsutani, Stefan problems with the unilateral boundary condition on the fixed boundary I. Osaka J. Math.,19 (1982), 365–403.

    MATH  MathSciNet  Google Scholar 

  33. S. Yotsutani, Stefan problems with the unilateral boundary condition on the fixed boundary II. Osaka J. Math.,20 (1983), 803–844.

    MATH  MathSciNet  Google Scholar 

  34. S. Yotsutani, Stefan problems with the unilateral boundary condition on the fixed boundary III. Osaka J. Math.,20 (1983), 845–862.

    MATH  MathSciNet  Google Scholar 

  35. S. Yotsutani, Stefan problems with the unilateral boundary condition on the fixed boundary IV. Osaka J. Math.,21 (1984), 149–167.

    MATH  MathSciNet  Google Scholar 

  36. A. Visintin, Sur le problème de Stefan avec flux non linéaire. Boll. Un. Mat. Ital. C, (5),18 (1981), 63–86.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hiroshima University, 730, Hiroshima, Japan

    Masayasu Mimura

  2. Department of Mathematics, Nagoya University, 464, Nagoya, Japan

    Yoshio Yamada

  3. Department of Applied Science, Miyazaki University, 880, Miyazaki, Japan

    Shoji Yotsutani

Authors
  1. Masayasu Mimura
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yoshio Yamada
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shoji Yotsutani
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Mimura, M., Yamada, Y. & Yotsutani, S. A free boundary problem in ecology. Japan J. Appl. Math. 2, 151–186 (1985). https://doi.org/10.1007/BF03167042

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03167042

Key words

Search

Navigation