Journal of Dynamics and Differential Equations

, Volume 27, Issue 3–4, pp 1027–1076 | Cite as

Diffusive Stability of Turing Patterns via Normal Forms



We investigate dynamics near Turing patterns in reaction–diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a “normal form” coordinate system near such Turing patterns which exhibits an approximate discrete conservation law. The key ingredients to the normal form is a conjugation of the reaction–diffusion system on the real line to a lattice dynamical system. At each lattice site, we decompose perturbations into neutral phase shifts and normal decaying components. As an application of our normal form construction, we prove nonlinear stability of Turing patterns with respect to perturbations that are small in \(L^1\cap L^\infty \), with sharp rates, recovering and slightly improving on results in Johnson and Zumbrun (Ann Inst H Poincaré Anal Non Linéaire 28:471–483, 2011) and Schneider (Commun Math Phys 178:679–702, 1996).


Reaction–diffusion systems Turing pattern Nonlinear stability Diffusive stability Normal form 



This work was partially supported by the National Science Foundation through Grant NSF-DMS-0806614.


  1. 1.
    Bricmont, J., Kupiainen, A.: Renormalization group and the Ginzburg–Landau equation. Commun. Math. Phys. 150, 193–208 (1992)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Bricmont, J., Kupiainen, A.: Stability of moving fronts in the Ginzburg–Landau equation. Commun. Math. Phys. 159, 287–318 (1994)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Deng, K., Levine, H.A.: The role of critical exponents in blow-up theorems: the sequel. J. Math. Anal. Appl. 243, 85–126 (2000)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Gallay, T., Scheel, A.: Diffusive stability of oscillations in reaction–diffusion systems. Trans. Am. Math. Soc. 363, 2571–2598 (2011)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)Google Scholar
  6. 6.
    Herrero, M.A., Velázquez, J.J.L.: Some results on blow up for semilinear parabolic problems. In: Degenerate Diffusions (Minneapolis, MN, 1991). IMA Volumes in Mathematics and its Applications, vol. 47, pp. 105–125. Springer, New York (1993)Google Scholar
  7. 7.
    Johnson, M.A.: Nonlinear stability of periodic traveling wave solutions of the generalized Korteweg–de Vries equation. SIAM J. Math. Anal. 41, 1921–1947 (2009)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Johnson, M.A., Zumbrun, K.: Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case. J. Differ. Equ. 249, 1213–1240 (2010)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Johnson, M.A., Zumbrun, K.: Nonlinear stability of periodic traveling-wave solutions of viscous conservation laws in dimensions one and two. SIAM J. Appl. Dyn. Syst. 10, 189–211 (2011)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Johnson, M.A., Zumbrun, K.: Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction–diffusion equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 471–483 (2011)Google Scholar
  11. 11.
    Johnson, M.A., Zumbrun, K., Noble, P.: Nonlinear stability of viscous roll waves. SIAM J. Math. Anal. 43, 577–611 (2011)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 16. Birkhäuser Verlag, Basel (1995)Google Scholar
  13. 13.
    Mielke, A.: Über maximale \(L^p\)-Regularität für Differentialgleichungen in Banach- und Hilbert-Räumen. Math. Ann. 277, 121–133 (1987)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Mielke, A.: Instability and stability of rolls in the Swift–Hohenberg equation. Commun. Math. Phys. 189, 829–853 (1997)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Murray, J.D.: Mathematical Biology. I: An Introduction. Interdisciplinary Applied Mathematics, vol. 17, 3rd edn. Springer, New York (2002)Google Scholar
  16. 16.
    Murray, J.D.: Mathematical Biology. II: Spatial Models and Biomedical Applications. Interdisciplinary Applied Mathematics, vol. 18, 3rd edn. Springer, New York (2003)Google Scholar
  17. 17.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978)Google Scholar
  18. 18.
    Sandstede, B., Scheel, A.: On the stability of periodic travelling waves with large spatial period. J. Differ. Equ. 172, 134–188 (2001)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Sandstede, B., Scheel, A., Schneider, G., Uecker, H.: Diffusive mixing of periodic wave trains in reaction–diffusion systems. J. Differ. Equ. 252, 3541–3574 (2012)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Scarpellini, B.: \({\cal{L}}^2\)-perturbations of periodic equilibria of reaction diffusion systems. NoDEA Nonlinear Differ. Equ. Appl. 1, 281–311 (1994)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Schneider, G.: Diffusive stability of spatial periodic solutions of the Swift–Hohenberg equation. Commun. Math. Phys. 178, 679–702 (1996)CrossRefMATHGoogle Scholar
  22. 22.
    Schneider, G.: Nonlinear stability of Taylor vortices in infinite cylinders. Arch. Ration. Mech. Anal. 144, 121–200 (1998)CrossRefMATHGoogle Scholar
  23. 23.
    Turing, A.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37–72 (1952)CrossRefGoogle Scholar
  24. 24.
    Uecker, H.: Diffusive stability of rolls in the two-dimensional real and complex Swift–Hohenberg equation. Commun. Partial Differ. Equ. 24, 2109–2146 (1999)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Zelik, S., Mielke, A.: Multi-pulse evolution and space–time chaos in dissipative systems. Mem. Am. Math. Soc. 198, vi+97 (2009)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

Personalised recommendations