Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-16T09:44:42.811Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady states of the reaction-diffusion equations. Part 1: Questions of existence and continuity of solution branches

Published online by Cambridge University Press:  17 February 2009

J. G. Burnell ,
A. A. Lacey  and
G. C. Wake
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

When material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant, the steady-state regime is governed by a coupled pair of nonlinear elliptic partial differential equations with linear boundary conditions. In this paper we consider questions of existence of solutions to these equations. It is shown that, with the exception of the special case in which the mass-transfer is uninhibited on the boundary, a solution always exists, whereas in this special case a solution exists only for sufficiently low values of the exothermicity. Bounds are established for the solutions and the occurrence of minimal and maximal solutions is shown for some cases. Finally the behaviour of the solution set with respect to one of the parameters is studied.

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 4 , April 1983 , pp. 374 - 391
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]Engelking, R., General topology (Polish Scientific Publishers, Warsaw, 1975).Google Scholar
[4]Fujita, H., “On the nonlinear equations δu + e u = 0 and v t = δv + e v”, Bull. Amer. Math. Soc. 75 (1969), 132–135.CrossRefGoogle Scholar
[5]Fujita, H., “On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations”, Proc. of Symposia in Pure Math. 28 Part I, Amer. Math. Soc. (1970), 105–113.CrossRefGoogle Scholar
[6]Keller, H. B. and Cohen, D. S., “Some positone problems suggested by nonlinear heat generation”, J. Math. Mech. 16 (1967), 1361–1376.Google Scholar
[7]Lacey, A. A., “Mathematical analysis of thermal runaway for spatially inhomogeneous reactions”, SIAM J. Appl. Math. (in press).Google Scholar
[8]Ladyzhenskaya, O. A. and Ural'tseva, N. N., Linear and quasilinear elliptic equations (Academic Press, New York, 1968).Google Scholar
[9]Lloyd, N. G., Degree theory (Cambridge University Press, 1978).Google Scholar
[10]Poore, A. B., “Stability and bifurcation phenomena in chemical reactor theory”, Arch. Rational Mech. Anal. 52 (1973), 358–388.CrossRefGoogle Scholar
[11]Sattinger, D. H., “Topics in stability and bifurcation theory”, Lecture Notes in Mathematics 309 (Springer-Verlag, New York, 1973).Google Scholar