# zbMATH — the first resource for mathematics

Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction-diffusion equations. (English) Zbl 1246.35034
The authors consider stability of periodic travelling waves (wavetrains) in systems of reaction diffusion equations. Under spectral stability assumptions they prove nonlinear stability in the sense that perturbations that are sufficiently small in $$L^1\cap H^k$$ decay in $$H^k$$ as $$t^{- 1/4}$$. Towards this they construct (by integral representations) suitable co-moving frames (which decay) that enjoy $$L^p$$ estimates.
The approach appears to provide an alternative and extension to convection of the classical results of G. Schneider.

##### MSC:
 35B35 Stability in context of PDEs 35K57 Reaction-diffusion equations 37L15 Stability problems for infinite-dimensional dissipative dynamical systems 35B10 Periodic solutions to PDEs 35C07 Traveling wave solutions 35K45 Initial value problems for second-order parabolic systems
##### Keywords:
Bloch decomposition; co-moving frames
Full Text:
##### References:
 [1] Doelman, A.; Sandstede, B.; Scheel, A.; Schneider, G., The dynamics of modulated wavetrains, Mem. amer. math. soc., 199, 934, (2009), viii+105 pp., ISBN 978-0-8218-4293-5 [2] Gardner, R., On the structure of the spectra of periodic traveling waves, J. math. pures appl., 72, 415-439, (1993) · Zbl 0831.35077 [3] Henry, D., Geometric theory of semilinear parabolic equations, Lecture notes in math., (1981), Springer-Verlag Berlin · Zbl 0456.35001 [4] Johnson, M.; Zumbrun, K., Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case, J. differential equations, 249, 5, 1213-1240, (2010) · Zbl 1198.35027 [5] Kato, T., Perturbation theory for linear operators, (1985), Springer-Verlag Berlin, Heidelberg [6] Mielke, A.; Schneider, G.; Uecker, H., Stability and diffusive dynamics on extended domains, (), 563-583 · Zbl 1004.35018 [7] Oh, M.; Zumbrun, K., Stability and asymptotic behavior of traveling-wave solutions of viscous conservation laws in several dimensions, Arch. ration. mech. anal., Arch. ration. mech. anal., 196, 1, 21-23, (2010), Erratum: · Zbl 1197.35075 [8] Pazy, A., Semigroups of linear operators and applications to partial differential equations, Appl. math. sci., ISBN: 0-387-90845-5, vol. 44, (1983), Springer-Verlag New York, Berlin, viii+279 pp · Zbl 0516.47023 [9] B. Sandstede, A. Scheel, G. Schneider, H. Uecker, Diffusive mixing of periodic wave trains in reaction-diffusion systems with different phases at infinity, draft, 2010. · Zbl 1298.35108 [10] Schneider, G., Nonlinear diffusive stability of spatially periodic solutions - abstract theorem and higher space dimensions, (), 159-167 · Zbl 0907.35015 [11] Uecker, H., Diffusive stability of rolls in the two-dimensional real and complex Swift-Hohenberg equation, Comm. partial differential equations, 24, 11-12, 2109-2146, (1999) · Zbl 0937.35135
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.