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Analysis on stability and non-existence of equilibrium for a general chemical reaction. (English) Zbl 1413.35262
Summary: This paper is concerned with a general chemical reaction model with respect to Neumann boundary condition. The stability of positive equilibrium and the non-existence of non-constant positive solution are discussed rigorously, respectively in Case 1: $$f(u)>0$$ and $$f_u(u)<0$$ for $$u>0$$ and Case 2: $$f(0)=0$$ and $$f_u(u)>0$$ for $$u>0$$. The techniques include the spectrum analysis of operators, the maximal principle, the upper and lower solution method and the implicit function theorem.
MSC:
 35K57 Reaction-diffusion equations 35B35 Stability in context of PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence
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