PHASE SINKS AND SOURCES AROUND TWO-DIMENSIONAL PERIODIC-WAVE SOLUTIONS OF REACTION-DIFFUSION-ADVECTION SYSTEMS
Résumé
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 hyperbolicparabolic 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.
Mots clés
periodic traveling-wave solutions reaction-diffusion systems asymptotic stability modulation systems dispersive estimates hyperbolic-parabolic systems AMS Subject Classifications: 35B35
35K57
35C07
35B40
35B10
37L15
periodic traveling-wave solutions
reaction-diffusion systems
asymptotic stability
modulation systems
dispersive estimates
hyperbolic-parabolic systems AMS Subject Classifications: 35B35
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