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The number of solutions to an equation from catalysis*

Published online by Cambridge University Press:  14 November 2011

S. P. Hastings  and
J. B. McLeod
S. P. Hastings
Affiliation:
SUNY Center at Buffalo, Buffalo, NY 14214, U.S.A.
J. B. McLeod
Affiliation:
The Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, England
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Synopsis

In catalysis theory there is interest in the number of solutions to the equation

with the boundary conditions

the parameters λ,β, γ being all positive and p a non-negative integer. The paper answers this question when γ is large, which is the interesting situation physically. Although the treatment is somewhat different in the cases p = 0 and p ≠ 0, the final answer is the same, that is, given β, there exist two positive functions λ1(γ) and λ2(γ) such that the problem has one solution if λ<λ1(γ), or λ>λ2(γ), three solutions if λ1(γ)<λ <λ2(γ), and two solutions if λ=λ1(γ) or λ=λ2(γ).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 101 , Issue 1-2 , 1985 , pp. 15 - 30
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Aris, R.. The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts (Oxford University Press, 1975).Google Scholar
2Dancer, E. N.. On the structure of solutions of an equation in catalysis theory when a parameter is large. J. Differential Equations 37 (1980), 404–437.CrossRefGoogle Scholar
3Parter, S. V.. Solutions of a differential equation arising in chemical reactor processes. SIAM J. Appl. Math. 26 (1974), 687–716.CrossRefGoogle Scholar
4Smoller, J. and Wasserman, A.. Global bifurcation of steady-state solutions. J. Differential Equations 39 (1981), 269–290.CrossRefGoogle Scholar
5Kapila, A. K. and Matkowsky, B. J.. Reactive-diffusive systems with Arrhenius kinetics: multiple solutions, ignition and extinction. SIAM. J. Appl. Math. 36 (1979), 373–389.CrossRefGoogle Scholar
6Kapila, A. K. and Matkowsky, B. J.. Reactive-diffusive systems with Arrhenius kinetics: the Robin problem. SIAM J. Appl. Math. 39 (1980), 391–401.CrossRefGoogle Scholar