Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics.

*(English)*Zbl 1371.35084Summary: In this paper, a reaction-diffusion system known as an autocatalytic reaction model is considered. The model is characterized by a system of two differential equations which describe a type of complex biochemical reaction. Firstly, some basic characterizations of steady-state solutions of the model are presented. And then, the stability of positive constant steady-state solution and the non-existence, existence of non-constant positive steady-state solutions are discussed. Meanwhile, the bifurcation solution which emanates from positive constant steady-state is investigated, and the global analysis to the system is given in one dimensional case. Finally, a few numerical examples are provided to illustrate some corresponding analytic results.

##### MSC:

35J57 | Boundary value problems for second-order elliptic systems |

35K57 | Reaction-diffusion equations |

92C45 | Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) |

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\textit{Y. Jia} et al., Discrete Contin. Dyn. Syst. 37, No. 9, 4785--4813 (2017; Zbl 1371.35084)

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