Existence and stability of travelling front solutions for general auto-catalytic chemical reaction systems (Q2843426)

The authors consider two component autocatalytic reaction-diffusion systems with nonlinearities \(\pm u f(v)\), with minus-case and constant diffusion \(d\) in the \(u\)-equation, and unit diffusion in the minus-case of the \(v\)-equation. The assumptions on \(f\) are as usual in this context that \(f(0)=0\), \(f(1)=1\) and positive \(f\) on the positive unit interval. The additional rather weak assumption in this paper is super-linear growth of \(f\) (and \(f\) being \(C^2\)-smooth). NEWLINENEWLINENEWLINEThe authors give an overview of known results, typcially for more special \(f\), and also review relevant results on the Fisher-KPP equation. The methods in this paper partly build on those in [\textit{A. Ghazaryan} et al., SIAM J. Math. Anal. 42, No. 6, 2434--2472 (2010; Zbl 1227.35057)], and the authors' previous work in [\textit{Y. Li} and \textit{Y. Wu}, SIAM J. Math. Anal. 44, No. 3, 1474--1521 (2012; Zbl 1259.35030)]. NEWLINENEWLINENEWLINEThe existence results of traveling front solutions on the one hand yield a selected family for \(d\) near \(1\) that decay exponentially and whose speeds limit to the minimum wave speed for the simpler case \(d=1\), including exponential decay. On the other hand, it is shown that the minimal wave speed is indeed a lower (upper) bound for \(0\leq d<1(d>1)\), and that there exists a unique wave front that decays algebraically. NEWLINENEWLINENEWLINEThe results on stability establish decay of sufficiently small integrable, uniformly continuous perturbations that also decay exponentially with certain weights (one of which may vanish). The decay is algebraic (exponential) in the supremum (exponentially weighted) norm.