Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction–diffusion equations
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 4, pp. 471-483.

Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling waves of systems of reaction–diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-speeds to which Schneiderʼs renormalization techniques do not appear to apply.

DOI : 10.1016/j.anihpc.2011.05.003
Mots-clés : Periodic traveling waves, Nonlinear stability, Bloch decomposition 
@article{AIHPC_2011__28_4_471_0,
     author = {Johnson, Mathew A. and Zumbrun, Kevin},
     title = {Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction{\textendash}diffusion equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {471--483},
     publisher = {Elsevier},
     volume = {28},
     number = {4},
     year = {2011},
     doi = {10.1016/j.anihpc.2011.05.003},
     mrnumber = {2823880},
     zbl = {1246.35034},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.05.003/}
}
TY  - JOUR
AU  - Johnson, Mathew A.
AU  - Zumbrun, Kevin
TI  - Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction–diffusion equations
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2011
SP  - 471
EP  - 483
VL  - 28
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2011.05.003/
DO  - 10.1016/j.anihpc.2011.05.003
LA  - en
ID  - AIHPC_2011__28_4_471_0
ER  - 
%0 Journal Article
%A Johnson, Mathew A.
%A Zumbrun, Kevin
%T Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction–diffusion equations
%J Annales de l'I.H.P. Analyse non linéaire
%D 2011
%P 471-483
%V 28
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2011.05.003/
%R 10.1016/j.anihpc.2011.05.003
%G en
%F AIHPC_2011__28_4_471_0
Johnson, Mathew A.; Zumbrun, Kevin. Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction–diffusion equations. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 4, pp. 471-483. doi : 10.1016/j.anihpc.2011.05.003. http://www.numdam.org/articles/10.1016/j.anihpc.2011.05.003/

[1] A. Doelman, B. Sandstede, A. Scheel, G. Schneider, The dynamics of modulated wavetrains, Mem. Amer. Math. Soc. 199 no. 934 (2009) | MR

[2] R. Gardner, On the structure of the spectra of periodic traveling waves, J. Math. Pures Appl. 72 (1993), 415-439 | MR | Zbl

[3] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Springer-Verlag, Berlin (1981) | MR | Zbl

[4] M. Johnson, K. Zumbrun, Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case, J. Differential Equations 249 no. 5 (2010), 1213-1240 | MR | Zbl

[5] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg (1985) | MR

[6] A. Mielke, G. Schneider, H. Uecker, Stability and diffusive dynamics on extended domains, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin (2001), 563-583 | MR | Zbl

[7] M. Oh, K. Zumbrun, Stability and asymptotic behavior of traveling-wave solutions of viscous conservation laws in several dimensions, Arch. Ration. Mech. Anal. 196 no. 1 (2010), 1-20, Arch. Ration. Mech. Anal. 196 no. 1 (2010), 21-23 | MR | Zbl

[8] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. vol. 44, Springer-Verlag, New York, Berlin (1983) | MR | Zbl

[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. | MR

[10] G. Schneider, Nonlinear diffusive stability of spatially periodic solutions – abstract theorem and higher space dimensions, Proceedings of the International Conference on Asymptotics in Nonlinear Diffusive Systems, Sendai, 1997, Tohoku Math. Publ. vol. 8, Tohoku Univ., Sendai (1998), 159-167 | MR | Zbl

[11] H. Uecker, Diffusive stability of rolls in the two-dimensional real and complex Swift–Hohenberg equation, Comm. Partial Differential Equations 24 no. 11–12 (1999), 2109-2146 | MR | Zbl

Cité par Sources :