Asymptotic behavior and positive solutions of a chemical reaction diffusion system.

*(English)*Zbl 0799.35122Summary: This paper is concerned with some qualitative analysis for a coupled system of three reaction diffusion equations which arises from certain chemical reactions first discovered by Belousov and Zhabotinskii. The analysis includes the existence of a bounded global time-dependent solution, the stability and instability of the zero solution, and the existence and nonexistence of a positive steady-state solution, including a global attractor of the system. The global existence and stability problem is determined by the method of upper and lower solutions, and the existence of a positive steady-state solution is based on the fixed point index and bifurcation theory. This analysis leads to a necessary and sufficient condition for the existence and nonexistence of a positive steady-state solution in relation to the various physical parameters of the system.

##### MSC:

35K57 | Reaction-diffusion equations |

92E20 | Classical flows, reactions, etc. in chemistry |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

35B32 | Bifurcations in context of PDEs |

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\textit{W. H. Ruan} and \textit{C. V. Pao}, J. Math. Anal. Appl. 169, No. 1, 157--178 (1992; Zbl 0799.35122)

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