Abstract
In this paper, we continue the mathematical study started in (Jang et al. in J. Dyn. Differ. Equ. 16:297-320, 2004; Ni and Tang in Trans. Am. Math. Soc. 357:3953-3969, 2005) on the analytic aspects of the Lengyel-Epstein reaction-diffusion system. First, we further analyze the fundamental properties of nonconstant positive solutions. On the other hand, we continue to consider the effect of the diffusion coefficient d. We obtain another nonexistence result for the case of large d by the implicit function theory, and investigate the direction of bifurcation solutions from $(u^{*},v^{*})$ . These results promote the Turing patterns arising from the Lengyel-Epstein reaction-diffusion system.
Highlights
1 Introduction In this paper, we continue to consider the Lengyel-Epstein reaction-diffusion system [, ]. This system captures the crucial feature of the CIMA reaction in an open unstirred gel reactor which gave the first experimental evidence of a Turing pattern in [ ]
The following theorem gives the nonexistence of nonconstant steady states to the problem ( . ) when d is not large
To obtain the nonexistence result for the case of large d, we first give the asymptotic behavior of positive solutions to ( . ) when d is sufficiently large
Summary
We continue to consider the Lengyel-Epstein reaction-diffusion system [ , ]. ) has a unique constant positive solution u∗, v∗ = α, + α , where α = a/ . In Section , we investigate the direction of bifurcation solutions from simple eigenvalue (i.e., the case dj = dk), which promotes the results in [ ].
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