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Steady states of the reaction-diffusion equations. Part II: Uniqueness of solutions and some special cases

Published online by Cambridge University Press:  17 February 2009

J. G. Burnell ,
A. A. Lacey  and
G. C. Wake
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Abstract

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In an earlier paper (Part I) the existence and some related properties of the solution to a coupled pair of nonlinear elliptic partial differential equations 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 consider the range of parameters for which the uniqueness of solution is assured and we also investigate the converse question of multiple solutions. The special case of the one dimensional shape of the infinite slab is investigated in full as this case admits to solution by integration.

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 4 , April 1983 , pp. 392 - 416
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Amann, H., “Fixed point equations and non-linear eigenvalue problems in ordered Banach spaces”, SIAM Rev. 18 (1976), 620–709.Google Scholar
[2]Aris, R., The mathematical theory of diffusion and reaction in permeable catalysts, Vol. 1 (Oxford University Press, 1975).Google Scholar
[3]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). 375–392.Google Scholar
[4]Keller, H. B. and Cohen, D. S., “Some positone problems suggested by nonlinear heat generation”, J. Math. Mech. 16 (1967), 1361–1376.Google Scholar
[5]Sattinger, D. H., “Topics in stability and bifurcation theory”, Lecture Notes in Mathematics 309 (Springer Verlag, New York, 1973).Google Scholar