2024/08/27 by Benjamin Melinand, Melinand, Benjamin, L. Miguel Rodrigues +1 · 1 citation
Computer Science · #35B10 #35B35 #35B40 #35C07 #35K57 #37L15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation
paper · pdf · doi:10.48550/arxiv.2408.14869
openalex publication_date 2024/08/27 · openalex created_date 2024/08/30 · openalex updated_date 2026/07/28
We develop a complete stability theory for two-dimensional periodic traveling waves of reaction-diffusion systems. More precisely, we identify a diffusive spectral stability assumption, prove that it implies nonlinear stability and provide a sharp asymptotic description of the dynamics resulting from both localized and critically nonlocalized perturbations. In particular, we show that the long-time behavior is governed at leading order by a second-order Whitham modulation system and elucidate how the intertwining of diffusive and dispersive effects may enhance decay rates. The latter requires a non trivial extension of the large-time estimates for constant-coefficient hyperbolic-parabolic operators to some classes of systems with no particular structure, including on one hand systems with a scalar-like - but not scalar - hyperbolic part and a cross-diffusion, and on the other hand anisotropic systems with dispersion.