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The solution via monotonicity methods of some nonscalar reaction-diffusion problems

Published online by Cambridge University Press:  20 January 2009

Manuel Delgado
Affiliation:
Universidad de Sevilla C/Tarfia S/N 41012 Sevilla, Spain
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Abstract

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We present some applications of monotonicity methods to the solution of certain nonscalar reaction-diffusion problems. In particular we prove existence under appropriate conditions and we introduce a convergent algorithm.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1996

References

REFERENCES

1. Amann, H., On the number of solutions of nonlinear equations in ordered Banach spaces, J. Fund. Anal. 11 (1972), 346384.CrossRefGoogle Scholar
2. Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620709.CrossRefGoogle Scholar
3. Blat, J. and Brown, K. J., Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Roy. Soc. Edinburgh 97A (1981), 2134.Google Scholar
4. Blat, J. and Brown, K. J., Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal. 17 (1986), 13391353.CrossRefGoogle Scholar
5. Cantrell, R. S. and Cosner, C., On steady-state problem for the Volterra-Lotka competition model with diffusion, Houston J. Math. 13 (1987), 337352.Google Scholar
6. Cosner, C. and Lazer, A. C., Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math. 44 (1984), 11121132.CrossRefGoogle Scholar
7. Dancer, E. N., On positive solutions of some pairs of differential equations, J. Differential Equations 60 (1985), 236258.CrossRefGoogle Scholar
8. Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order (Springer-Verlag, New York 1977).CrossRefGoogle Scholar
9. Hernandez, J., Maximum principles and decoupling for positive solutions of reaction-diffusion systems, in Reaction-diffusion Equations (Brown, K. J. and Lacey, A. A. eds., Clarendon Press, Oxford 1990).Google Scholar
10. Li, L., Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc. 305 (1988), 143166.CrossRefGoogle Scholar
11. Pao, C. V., On nonlinear reaction-diffusion systems, J. Math. Anal. Appl. 87 (1982), 165198.CrossRefGoogle Scholar
12. Adams, R., Sobolov spaces (Academic Press, New York, 1975).Google Scholar
13. Gilbarg, D. and Hörmander, L., Intermediate Schauder estimates, Arch. Rational Mech. Anal. 74 (1980), 297318.CrossRefGoogle Scholar