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Uniqueness, multiplicity and stability for positive solutions of a pair of reaction–diffusion equations

Published online by Cambridge University Press:  14 November 2011

Yihong Du
Yihong Du
Affiliation:
Department of Mathematics, Statistics and Computing Science, University of New England, Armidale, NSW 2351, Australia
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Abstract

We study the number and stability of the positive solutions of a reaction–diffusion equation pair. When certain parameters in the equations are large, the equation pair can be viewed as singular or regular perturbations of some single (or essentially single) equation problems, for which the number and stability of their solutions can be well understood. With the help of these simpler equations, we are able to obtain a rather complete understanding of the number and stability of the positive solutions for the equation pair for the cases that certain parameters are large. In particular, we obtain a fairly satisfactory description of the positive solution set of the equation pair.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 4 , 1996 , pp. 777 - 809
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Amann, H.. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976), 620–709.CrossRefGoogle Scholar
2Bonilla, L. L. and Verlarde, M. G.. Singular perturbation approach to the limit cycle and global patterns in a nonlinear diffusion–reaction problem with autocatalysis and saturation law. J. Math. Phys. 20(1979), 2692–703.CrossRefGoogle Scholar
3Brezis, H. and Turner, R. E. L.. On a class of superlinear elliptic problems. Comm. Partial Differential Equations 2 (1977), 601–14.CrossRefGoogle Scholar
4Chow, S. N. and Hale, J. K.. Methods of Bifurcation Theory (Berlin: Springer, 1982).CrossRefGoogle Scholar
5Crandall, M. and Rabinowitz, P. H.. Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rational Mech. Anal. 52 (1973), 161–80.CrossRefGoogle Scholar
6Dancer, E. N.. On the indices of fixed points of mappings in comes and applications. J. Math. Anal. Appl. 91 (1983), 131–51.CrossRefGoogle Scholar
7Dancer, E. N.. The effect of domain shape on the number of positive solutions of certain nonlinear equations. J. Differential Equations 74 (1988), 120–56.CrossRefGoogle Scholar
8Dancer, E. N.. On the existence and uniqueness of positive solutions for competing species models with diffusion. Trans. Amer. Math. Soc. 36 (1991), 829–59.CrossRefGoogle Scholar
9Dancer, E. N.. On uniqueness and stability for solutions of singularly perturbed predator–prey type equations with diffusion. J. Differential Equations 102 (1993), 1–32.CrossRefGoogle Scholar
10Dancer, E. N. and Du, Y.. Competing species equations with diffusion, large interactions and jumping nonlinearities. J. Differential Equations 114 (1994), 434–75.CrossRefGoogle Scholar
11Dancer, E. N. and Guo, Z. M.. Uniqueness and stability for solutions of competing species equations with large interactions. Comm. Appl. Nonlinear Analysis 1 (1994), 19–45.Google Scholar
12Deimling, K.. Nonlinear Functional Analysis (Berlin: Springer, 1985).CrossRefGoogle Scholar
13Du, Y.. Fixed points of increasing operators in ordered Banach spaces and applications. Appl. Anal. 38(1990), 1–20.CrossRefGoogle Scholar
14Gilbarg, D. and Trudinger, N.. Elliptic Partial Differential Equations of Second Order (Berlin: Springer, 1977).CrossRefGoogle Scholar
15Gossez, J. P. and Dozo, E. Lami. On the principal eigenvalue of a second order linear elliptic problem. Arch. Rational Mech. Anal. 89 (1985), 169–75.CrossRefGoogle Scholar
16Henry, D.. Geometric Theory of Semilinear Parabolic Equations (Berlin: Springer, 1981).CrossRefGoogle Scholar
17Ibanez, J. L. and Verlarde, M. G.. Multiple steady states in a simple reaction–diffusion model with Michaelis–Menten (first-order Hinshelwood–Langmuir) saturation law: the limit of large separation in the two diffusion constants. J. Math. Phys. 19 (1978), 151–6.CrossRefGoogle Scholar
18Kato, T.. Perturbation Theory for Linear Operators (Berlin: Springer, 1966).Google Scholar
19Krasnoselskii, M. A.. Positive Solutions of Operator Equations (Groningen: Noordhoff, 1964).Google Scholar
20Nussbaum, R. D.. The fixed point index for locally condensing maps. Ann. Mat. Pura Appl. 87 (1971), 217–58.CrossRefGoogle Scholar
21Protter, M. and Weinberger, H.. Maximum Principles in Differential Equations (Englewood Cliffs, N.J.: Prentice-Hall, 1967).Google Scholar
22Ruan, W.. Asymptotic behavior and positive steady-state solutions of a reaction–diffusion model with autocatalysis and saturation law. Nonlinear Anal. 21 (1993), 439–56.CrossRefGoogle Scholar
23Sattinger, D. H.. Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics 309 (Berlin: Springer, 1973).CrossRefGoogle Scholar
24Saut, J. and Teman, R.. Generic properties of nonlinear boundary value problems. Comm. Partial Differential Equations 4 (1979), 293–319.CrossRefGoogle Scholar