Skip to main content
SpringerLink
Log in

A criterion for exponential stability of the zero solution to a weakly-coupled cooperation-diffusion parabolic system

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

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.

References

  1. B. P. Demidovich, Lectures on the Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  2. Yu. L. Daletskiî and M. G. Kreîn, Stability of Solutions to Differential Equations [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  3. T. I. Zelenyak, “On stabilization of solutions to boundary value problems for parabolic equations with a single spatial variable,” Differentsial'nye Uravneniya,4, No. 1, 34–45 (1968).

    Google Scholar 

  4. H. Kielhoffer, “On the Lyapunov stability of stationary solutions of semilinear parabolic differential equations,” J. Differential Equations,22, No. 1, 193–208 (1976).

    Google Scholar 

  5. V. S. Belonosov and M. P. Vishnevskiî, “On stability of stationary solutions to nonlinear parabolic systems,” Mat. Sb.,104, No. 4, 535–558 (1977).

    Google Scholar 

  6. V. S. Belonosov, “Estimates for solutions to parabolic systems in weighted Hölder classes and their applications,” Mat. Sb.,110, No. 2, 163–188 (1979).

    Google Scholar 

  7. M. P. Vishnevskiî, “Periodic and close-to-periodic solutions to nonlinear parabolic systems,” Sibirsk. Mat. Zh.,22, No. 6, 208–209 (1981).

    Google Scholar 

  8. M. P. Vishnevskiî, “Integral sets of nonlinear parabolic systems,” Dinamika Sploshn. Sredy,54, 74–84 (1982).

    Google Scholar 

  9. M. P. Vishnevskiî, “Invariant sets of nonlinear parabolic systems,” in: Some Applications of Functional Analysis to Equations of Mathematical Physics [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk, 1986, pp. 32–56.

    Google Scholar 

  10. A. Lazer, “Some remarks on periodic solutions of parabolic differential equations,” in: Dynamical Systems. I (Eds. Bednarek Cesari), Acad. Press New York, 1982, pp. 25–46.

    Google Scholar 

  11. E. Dancer and P. Hess, “On the stable solutions of quasilinear periodic-parabolic problems,” Ann. Scuola Norm. Sup. Pisa,14, 123–141 (1987).

    Google Scholar 

  12. T. I. Zelenyak, Qualitative Theory of Boundary Value Problems for Quasilinear Parabolic Equations [in Russian], Novosibirsk Univ., Novosibirsk (1972).

    Google Scholar 

  13. T. I. Zelenyak, “On qualitative properties of solutions to mixed quasilinear problems of parabolic type,” Mat. Sb.,104, No. 3, 486–510 (1977).

    Google Scholar 

  14. M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prinston-Hall, New York (1967).

    Google Scholar 

  15. M. P. Vishnevskiî, “A criterion for stability of solutions to mixed problems for parabolic equations,” Trudy Sem. Sobolev, Novosibirsk,1, 5–23 (1984).

    Google Scholar 

  16. R. Costen and C. Holland, “Instability results for reaction-diffusion equations with Neumann boundary conditions,” J. Differential Equations,27, 266–273 (1978).

    Google Scholar 

  17. H. Matano, “Asymptotic behavior and stability of solutions of semilinear diffusion equation,” Publ. Res. Inst. Math. Sci.,15, 401–454 (1979).

    Google Scholar 

  18. K. Kishimoto and H. Weinberger, “The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains,” J. Differential Equations,58, 15–21 (1985).

    Google Scholar 

  19. P. Hess, “Spatial homogeneity of stable solutions of some periodic-parabolic problems with Neumann boundary conditions,” J. Differential Equations,68, 320–331 (1987).

    Google Scholar 

  20. P. Hess and T. Kato, “On some linear and nonlinear eigenvalue problems with an indefinite weight function,” Comm. Partial Differential Equations,5, 999–1030 (1980).

    Google Scholar 

  21. A. Beltramo and P. Hess, “On the principal eigenvalue of a periodic-parabolic operators,” Comm. Partial Differential Equations,9, 919–941 (1984).

    Google Scholar 

  22. P. Hess, “On the eigenvalue problem for weakly-coupled elliptic systems,” Arch. Rational Mech. Anal.,81, 151–159 (1983).

    Google Scholar 

  23. R. Cantrell and K. Schmitt, “On the eigenvalue problem for coupled elliptic system,” SIAM J. Math. Anal.,17, No. 4, 850–862 (1986).

    Google Scholar 

  24. L. Payne, “Some remarks on maximum principles,” J. Analyse Math.,30, 421–433 (1976).

    Google Scholar 

  25. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  26. A. Friedman, Partial Differential Equations of Parabolic Type [Russian translation], Mir, Moscow (1968).

    Google Scholar 

Download references

Authors
  1. M. P. Vishnevskiî
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 3, pp. 27–42, May–June, 1993.

Translated by G. V. Dyatlov

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vishnevskiî, M.P. A criterion for exponential stability of the zero solution to a weakly-coupled cooperation-diffusion parabolic system. Sib Math J 34, 419–432 (1993). https://doi.org/10.1007/BF00971217

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00971217

Keywords

Search

Navigation