Instability of planar traveling waves in bistable reaction-diffusion systems
Masaharu Taniguchi · Discrete and Continuous Dynamical Systems - B · 2003
This paper is concerned with the stability of a planar travelingwave in a cylindrical domain. The equation describesactivator-inhibitor systems in chemistry or biology. The wave hasa thin transition layer and is constructed by singularperturbation methods. Let $\varepsilon$ be the width of the layer. We showthat, if the cross section of the domain is narrow enough, thetraveling wave is asymptotically stable, while it is unstable ifthe cross section is wide enough by studying the linearizedeigenvalue problem. For the latter case, we study the wavelengthassociated with an eigenvalue with the largest real part, which iscalled the fastest growing wavelength. We prove that thiswavelength is $O(\varepsilon^{1/3})$ as $\varepsilon$ goes to zeromathematically rigorously. This fact shows that, if unstableplanar waves are perturbed randomly, this fastest growingwavelength is selectively amplified with as time goes on. For thisanalysis, we use a new uniform convergence theorem for someinverse operator and carry out the Lyapunov-Schmidt reduction.