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Steady states of the reaction-diffusion equations. Part III: Questions of multiplicity and uiqueness of solutions

Published online by Cambridge University Press:  17 February 2009

J. G. Burnell ,
A. A. Lacey  and
G. C. Wake
J. G. Burnell
Affiliation:
Mathematics Department, Victoria University, Private Bag, Wellington, New Zealand. Please address all correspondence to Dr. Wake at this address.
A. A. Lacey
Affiliation:
Mathematics Department, Heriot-Watt University, Edinburgh EH14 4AS, Scotland.
G. C. Wake
Affiliation:
Mathematics Department, Victoria University, Private Bag, Wellington, New Zealand. Please address all correspondence to Dr. Wake at this address.
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Abstract

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In earlier papers (Parts I and II) existence and uniqueness of the solutions to a coupled pair of nonlinear elliptic partial differential equations with linear boundary conditions was considered. These equations arise when material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant. In this paper we establish the existence of multiple solutions for many different values of the parameters not considered in the earlier parts. It is shown that the case, also omitted in earlier parts, with perfect thermal and mass transfer on the boundary (the double-Dirichlet case) does have a unique solution for sufficiently large values of the exothermicity or an equivalent parameter. The methods of solution provide specific bounds on the region of existence of multiple solutions.

Type
Research Article
Information
The ANZIAM Journal , Volume 27 , Issue 1 , July 1985 , pp. 88 - 110
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Burnell, J. G., Lacey, A. A. and Wake, G. C., “Steady states of the reaction-diffusion equations. Part I: Questions of existence and continuity of solution branches”, J. Austral. Math. Soc. Ser. B 24 (1983), 374–391.CrossRefGoogle Scholar
[2]Burnell, J. G., Lacey, A. A. and Wake, G. C., “Steady states of the reaction-diffusion equations. Part II: Uniqueness of solutions and some special cases”, J. Austral. Math. Soc. Ser. B 24 (1983), 392–416.CrossRefGoogle Scholar
[3]Keller, H. B. and Cohen, D. S., “Some positone problems suggested by nonlinear heat generation”, J. Math. Mech. 16 (1967), 1361–1376.Google Scholar