On the existence and stability of spatially structured solutions of the reaction-diffusion equations

J. G. Graham-eagle*, B. F. Gray, G. C. Wake

*Corresponding author for this work

Research output: Contribution to journalArticle

2 Citations (Scopus)


The reaction-diffusion equations for the well-known 'Brusselator' chemical kinetic model are investigated when the model is made consistent with the principle of detailed balance. In contrast to the original model, the corrected one does not show solutions with 'spatial structure' i.e. solutions with multiple internal maxima and multiple internal global minima in both dependent variables. Sufficient conditions for stability of the solutions are given in terms of a Rayleigh quotient for general boundary conditions for nonlinear reaction-diffusion equations in general. For the particular case of the 'Brusselator' it is shown that conditions for a change of stability are much more unlikely to be attained as a result of the restrictions of detailed balancing. The detailed sufficiency condition for the stability of any steady-state solution and for no branching from the 'equilibrium' branch solution depends on whether the solution has global extrema inside the region, on its boundary, or both.

Original languageEnglish
Pages (from-to)25-35
Number of pages11
JournalIMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Issue number1
Publication statusPublished - 1986
Externally publishedYes


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