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Bifurcation of Solutions to Reaction-Diffusion Systems with Jumping Nonlinearities

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Abstract

Bifurcation of stationary solutions to reaction-diffusion systems of activator-inhibitor type with jumping nonlinearities are located. The result can be understood as a certain destabilizing effect of jumping terms.

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References

  1. Dancer E.N. (1974). On the structure of solutions of non-linear eigenvalue problems. Indiana Univ. Math. J., 23:1069–1076.

    Article  MathSciNet  MATH  Google Scholar 

  2. Drabek P. and Kučera M. (1988). Reaction-diffusion systems: Destabilizing effect of unilateral conditions. Nonlinear Anal., 12:1173–1192.

    MathSciNet  Google Scholar 

  3. Eisner J. and Kučera M (1997). Spatial patterns for reaction-diffusion systems with conditions described by inclusions. Appl. Math., 42:421–449.

    Article  MathSciNet  Google Scholar 

  4. Eisner J. (1996). Critical and bifurcation points of reaction-diffusion systems with conditions given by inclusions. Preprint Math. Inst. Acad. Sci. Czech Rep., 118. Submitted to Nonlinear Anal.

    Google Scholar 

  5. Fučik S and Kufner A. (1980). Nonlinear Differential Equations. Elsevier, Amsterdam.

    Google Scholar 

  6. Kielhöfer H. (1974). Stability and semilinear evolution equations in Hilbert space. Arch. Rational Mech. Anal., 57:150–165.

    MATH  Google Scholar 

  7. Kučera M (1982). Bifurcation points of variational inequalities. Czechoslovak Math. J., 32:208–226.

    MathSciNet  Google Scholar 

  8. Kučera M and Bosák M. (1993). Bifurcation for quasi-variational inequalities of reaction-diffusion type. SAACM, 3:111–127.

    Google Scholar 

  9. Kučera M (1995). Bifurcation of solutions to reaction-diffusion systems with unilateral conditions. In Navier-Stokes Equations and related Topics (Edited by A. Sequeira), pp. 307–322. Plenum Press, New York.

    Google Scholar 

  10. Kučera M (1996). Reaction-diffusion systems: Bifurcation and stabilizing effect of conditions given by inclusions. Nonlinear Anal., 27:249–260.

    MathSciNet  Google Scholar 

  11. Kučera M (1997). Bifurcation of solutions to reaction-diffusion systems with conditions described by inequalities and inclusions. Nonlinear Anal., 30:3683–3694.

    MathSciNet  Google Scholar 

  12. Kučera M (1997). Reaction-diffusion systems: Stabilizing effect of conditions described by quasivariational inequalities. Czechoslovak Math. J., 47:469–486.

    MathSciNet  Google Scholar 

  13. Lions J. L. (1969). Quelques méthodes de resolution de problémes aux limites non linéaires. Paris.

    Google Scholar 

  14. Mimura M., Nishiura Y. and Yamaguti M. (1979). Some diffusive prey and predator systems and their bifurcation problems. Ann. New York Acad. Sci., 316:490–521.

    MathSciNet  Google Scholar 

  15. Murray J. D. (1993). Mathematical Biology. Biomathematics Texts, Vol. 19. Springer-Verlag Berlin, Heidelberg.

    Google Scholar 

  16. Nishiura Y. (1982). Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Anal, 13:555–593.

    Article  MathSciNet  MATH  Google Scholar 

  17. Nirenberg L. (1974). Topics in Nonlinear Functional Analysis. Academic Press, New York.

    Google Scholar 

  18. Quittner P. (1987). Bifurcation points and eigenvalues of inequalities of reaction-diffusion type. J. Reine Angew. Math., 380:1–13.

    MathSciNet  MATH  Google Scholar 

  19. Rabinowitz P.H. (1971). Some global results for nonlinear eigenvalue problems. J. Funct. Anal, 7:487–513.

    Article  MathSciNet  MATH  Google Scholar 

  20. Sattinger D.H. (1973). Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics 309. Springer-Verlag, Berlin-Heidelberg-New York.

    Google Scholar 

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© 2002 Kluwer Academic Publishers

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Eisner, J., Kučera, M. (2002). Bifurcation of Solutions to Reaction-Diffusion Systems with Jumping Nonlinearities. In: Sequeira, A., da Veiga, H.B., Videman, J.H. (eds) Applied Nonlinear Analysis. Springer, Boston, MA. https://doi.org/10.1007/0-306-47096-9_7

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  • DOI: https://doi.org/10.1007/0-306-47096-9_7

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-306-46303-7

  • Online ISBN: 978-0-306-47096-7

  • eBook Packages: Springer Book Archive

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