Comments to ‘Stability of traveling waves for deterministic and stochastic delayed reaction-diffusion equation based on phase shift’

Abstract

In Liu et al. (2023), the authors set out to prove orbital stability for traveling waves in a (stochastic) delay reaction–diffusion equation. For the deterministic proof, they follow a phase-tracking technique as developed by Zumbrun and Howard (1998) [1] and for the stochastic version, they use a stochastic phase-tracking technique as developed by Hamster and Hupkes (2019, 2020a,b) [2,3] (which in turn is based on the aforementioned technique by Howard and Zumbrun). Note that the authors do not acknowledge the origins of their techniques.In the deterministic case, stability results are known (Mei, 2009) [4], but the application of phase-tracking to this problem appears to be new. However, there is a very subtle interplay between the time-dependent phase and the delay in the delay equation which is not accounted for, nor is explained or proved if the chosen phase has the same properties in the general framework of delay semigroups as for regular semigroups.The proof of the stochastic case of course has the same issues, but let us assume that the issues can be resolved, then there are still several issues with the proof. Especially, the phase tracking introduces terms that depend on the gradient of the solution which must be bounded in the H1 norm. This introduces singularities in the bounds that do not show in these proofs.