Skip to main content
SpringerLink
Log in

Existence of fast traveling waves for some parabolic equations: A dynamical systems approach

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

We study semilinear elliptic equationsδu + cu x =f(u,∇u) andδ 2 u + cu x =f(u,∇u,∇ 2 u) in infinite cylinders (x,y) ∃ ℝ×Ω⊂ℝ n+1 using methods from dynamical systems theory. We construct invariant manifolds, which contain the set of bounded solutions and then study a singular limitc→∞, where the equations change type from elliptic to parabolic. In particular we show that on the invariant manifolds, the elliptic equation generates a smooth dynamical system, which converges to the dynamical system generated by the parabolic limit equation. Our results imply the existence of fast traveling waves for equations like a viscous reactive 2d-Burgers equation or the Cahn-Hillard equation in infinite strips.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Angenent, The Morse-Smale property for a semilinear parabolic equation.J. Diff. Eq. 62 (1986), 427–442.

    Google Scholar 

  2. S. Angenent and B. Fiedler, The dynamics of rotating waves in scalar reaction-diffusion equations.Trans. Am. Math. Soc. 307 (1988), 545–568.

    Google Scholar 

  3. J. Appell and P. Zabrejko,Nonlinear Superposition Operators, Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  4. P. Bates and P. Fife, Spectral comparison principles for the Cahn-Hillard and phase-field equations, and time scales for coarsening.Physica D 43 (1990), 335–348.

    Google Scholar 

  5. H. Berestycki and L. Nirenberg,Some Qualitative Properties of Solutions of Semilinear Elliptic Equations in Cylindrical Domains, Coll. Anal. etc. Academic Press, Boston, 1990, pp. 115–164.

    Google Scholar 

  6. P. Brunovský, X. Mora, P. Poláčik, and J. Solà-Morales, Asymptotic behavior of solutions of semilinear elliptic equations on an unbounded strip.Acta Math. Univ. Comenian. 60 (1991), 163–183.

    Google Scholar 

  7. J. Cahn, On spinodal decomposition.Acta Metallurg. 9 (1961), 795–801.

    Google Scholar 

  8. J. Cahn and J. Hillard, Free energy of a nonuniform system. I. Interfacial free energy.J. Chem. Phys. 28 (1958), 258–267.

    Google Scholar 

  9. A. Calsina, X. Mora, and J. Solà-Morales, The dynamical approach to elliptic problems in cylindrical domains and a study of their parabolic singular limit.J. Diff. Eq. 102 (1993), 244–304.

    Google Scholar 

  10. A. Calsina, J. Solà-Morales and M. València, Bounded solutions of some nonlinear elliptic equations in cylindrical domains. Preprint, Barcelona, 1993.

  11. J. Carr, M. Gurtir, and M. Slemrod, Structured phase transition on a finite interval.Arch. Rat. Mech. Anal. 86 (1984), 317–351.

    Google Scholar 

  12. S.-N. Chow and K. Lu,C k centre unstable manifolds.Proc. Roy. Soc. Edinburgh 108A (1988), 303–320.

    Google Scholar 

  13. P. Constantin, C. Foias, B. Nicolaenko, and R. Temam,Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Appl. Math. Sci. 70, Springer-Verlag, New York, 1989.

    Google Scholar 

  14. B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations.J. Diff. Eq. 125 (1996), 239–281.

    Google Scholar 

  15. G. Fischer, Zentrumsmannigfaltigkeiten bei elliptischen Differentialgleichungen.Math. Nachr. 115 (1984), 137–157.

    Google Scholar 

  16. J. Hale,Asymptotic Behavior of Dissipative Systems, Math. Surv. Monogr. 25, AMS, Providence, RI, 1988.

    Google Scholar 

  17. J. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation.J. Diff. Eq. 73 (1988), 197–214.

    Google Scholar 

  18. S. Heinze,Travelling Waves for Semilinear Parabolic Partial Differential Equations in Cylindrical Domains, Thesis, Heidelberg, 1989.

  19. D. Henry,Geometric Theory of Semiliniar Parabolic Equations, Lect. Notes Math. 840, Springer-Verlag, New York, 1981.

    Google Scholar 

  20. D. Henry, Some infinite dimensional Morse-Smale systems defined by parabolic equations.J. Diff. Eq. 59 (1985), 165–205.

    Google Scholar 

  21. M. Hirsch, C. Pugh, and M. Shub,Invariant Manifolds, Lect. Notes Math. 583, Springer-Verlag, New York, 1976.

    Google Scholar 

  22. T. Kato,Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.

    Google Scholar 

  23. K. Kirchgässner, Wave solutions of reversible systems and applications.J. Diff. Eq. 45 (1982), 113–127.

    Google Scholar 

  24. M. Kwak, Finite dimensional description of convective reaction-diffusion equations.J. Dyn. Diff. Eq. 4 (1992).

  25. M. Kwak, Finite dimensional inertial forms for the 2D Navier-Stokes equations.Indiana Univ. Math. J. 41 (1992), 927–981.

    Google Scholar 

  26. J. Mallet-Paret and G. Sell,The Principle of Spatial Averaging and Inertial Manifolds for Reaction-Diffusion Equations, Coll. Nonlin. Semigroups, Part. Diff. Eq. Attract.,Lect. Notes Math. 1248, Springer-Verlag New York, 1987.

    Google Scholar 

  27. A. Mielke, A reduction principle for nonautonomous systems in infinitedimensional spaces.J. Diff. Eq. 65 (1986), 68–88.

    Google Scholar 

  28. A. Mielke, Essential manifolds for elliptic problems in infinite cylinders. Preprint (1990).

  29. A. Mielke, Essential manifolds for an elliptic problem in an infinite strip.J. Diff. Eq. 110 (1994), 322–355.

    Google Scholar 

  30. K. Mischaikow, Global asymptotic dynamics of gradient-like bistable equations. Preprint CDSNS 92–94.

  31. X. Mora, Finite-dimensional attracting invariant manifolds for damped semilinear wave equations.Res. Notes Math. 155 (1987), 172–183.

    Google Scholar 

  32. X. Mora and J. Solà-Morales, Existence and non-existence of finite-dimensional globally attracting invariant manifolds in semilinear damped wave equations. InDynamics of Infinite Dimensional Systems, Springer-Verlag, New York, 1987, pp. 187–210.

    Google Scholar 

  33. X. Mora and J. Solà-Morales, The singular limit dynamics of semilinear damped wave equations.J. Diff. Eq. 78 (1989), 262–307.

    Google Scholar 

  34. N. Nadirashvili, Rotating waves and the Morse-Smale property for reaction-diffusion equations. Preprint

  35. A. Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag New York, 1985.

    Google Scholar 

  36. A. Scheel, Fast travelling waves of convective reaction diffusion equations.C.R. Acad. Sci. Paris, Ser. I 317 (1993), 347–351.

    Google Scholar 

  37. A. Scheel, Fast travelling waves for the Cahn-Hillard equation in an infinite strip.C.R. Acad. Sci. Paris Ser. I 317 (1993), 261–265.

    Google Scholar 

  38. M. Shub,Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987.

    Google Scholar 

  39. G. Sivashinsky, On flame propagation under conditions of stoichiometry.SIAM J. Appl. Math. 39 (1980), 67–82.

    Google Scholar 

  40. R. Temam,Infinite Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer-Verlag New York, 1988.

    Google Scholar 

  41. H. Triebel,Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, 1978.

    Google Scholar 

  42. A. Vanderbauwhede, Center manifolds, normal forms and elementary bifurcations.Dynamic. Report. 2 (1989), 89–169.

    Google Scholar 

  43. A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions.Dynamic. Report. 1 (1992), 125–163.

    Google Scholar 

  44. A. Vanderbauwhede and S. Van Gils, Center manifolds and contractions on a scale of Banach spaces.J. Funct. Anal. 72 (1987), 209–224.

    Google Scholar 

  45. K. Yoshida,Functional Analysis, Springer-Verlag, New York, 1965.

    Google Scholar 

  46. S. Zheng, Asymptotic behavior of solutions to the Cahn-Hillard equation.Appl. Anal. 23 (1986) 165–184.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik I, Freie Universität Berlin, Arnimallee 2-6, D-14195, Berlin, Germany

    Arnd Scheel

Authors
  1. Arnd Scheel
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Scheel, A. Existence of fast traveling waves for some parabolic equations: A dynamical systems approach. J Dyn Diff Equat 8, 469–547 (1996). https://doi.org/10.1007/BF02218843

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02218843

Key words

Search

Navigation