Stability of Traveling Front Solutions with Algebraic Spatial Decay for Some Autocatalytic Chemical Reaction Systems

Abstract

This paper is concerned with the linear and nonlinear asymptotic stability of traveling front solutions with algebraic spatial decay for some autocatalytic chemical reaction systems, especially for the typical autocatalytic system with reaction rate $u^qv^p$ when $p>1$ and $q\ge 1$. First, for the autocatalytic systems with equal diffusion rates and with more general initial values, the wave fronts with noncritical speeds are proved to be nonlinearly asymptotically stable in some special polynomially weighted spaces for the case where $p>1$ and $q=1$, and the waves fronts with noncritical speeds are proved to be linearly asymptotically stable for the case where $p>1$ and $q>1$. Second, by applying special transformations and appropriate matrix perturbation theories, we establish some abstract results on the existence and analyticity of the Evans function for the more general linear ODE systems with slow algebraic decaying coefficients. Third, by detailed spectral estimates and by applying our abstract results on the Evans function to the autocatalytic systems for the case where $p>1$ and $q\ge 1$ and when the two diffusion rates are close, we prove that the wave fronts with noncritical speeds are linearly exponentially stable in some exponentially weighted spaces. Finally, for the autocatalytic systems with $p>1$ and $q=1$ it can be shown that if the initial perturbation of the wave in $C_{\rm unif}(R)$ space is small in both the unweighted norm and the exponentially weighted norm, then the perturbation stays small in the unweighted norm and decays exponentially in the exponentially weighted norm; further, we can prove that if the initial perturbation is, in addition, small in the $L^1$ norm, then the perturbation also stays small in the $L_1(R)$ norm and decays algebraically in the $C_{\rm unif}(R)$ norm.