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The Fine Dynamics of the Chafee–Infante Equation

Part of the Lecture Notes in Mathematics book series (LNM,volume 2085)

Abstract

In this chapter, we introduce the deterministic Chafee–Infante equation. This equation has been the subject of intense research and is very well understood now. We recall some properties of its longtime dynamics and in particular the structure of its attractor. We then define reduced domains of attraction that will be fundamental in our study and give a result describing precisely the time that a solution starting form a reduced domain of attraction needs to reach a stable equilibrium. This result is then proved using the detailed knowledge of the attractor and classical tools such as the stable and unstable manifolds in a neighborhood of an equilibrium.

Keywords

  • Unstable Manifold
  • Global Attractor
  • Deterministic System
  • Heteroclinic Orbit
  • Deterministic Solution

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. H. Brezis, Functional Analysis (Masson, Paris, 1983)

    Google Scholar 

  2. I.D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dynamical Systems (Acta Scientific Publishing House, Kharkov, 2002)

    Google Scholar 

  3. N. Chafee, E.F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type. Appl. Anal. 4, 17–37 (1974)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. A. Eden, C. Foias, B. Nicolaenko, R. Temam, Exponential Attractors for Dissipative Evolution Equations (Wiley/Massons, New York/Paris, 1994)

    MATH  Google Scholar 

  5. W.G. Faris, G. Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise. J. Phys. A 10, 3025–3055 (1982)

    CrossRef  MathSciNet  Google Scholar 

  6. B. Fiedler, A. Scheel, Spatio-temporal dynamics of reaction-diffusion patterns. J. Phys. A Math. Gen. 15, 3025–3055 (1982)

    CrossRef  Google Scholar 

  7. J.K. Hale, in Infinite-Dimensional Dynamical Systems. Geometric Dynamics (Rio de Janeiro, 1981). Lecture Notes in Mathematics, vol. 1007 (Springer, Berlin, 1983)

    Google Scholar 

  8. D. Henry, Geometric Theory of Semilinear Parabolic Equations (Springer, Berlin, 1983)

    Google Scholar 

  9. D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations. J. Differ. Equat. 59, 165–205 (1985)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. G. Raugel, Global attractors in partial differential equations, in Handbook of Dynamical Systems, vol. 2, ed. by B. Fiedler (Elsevier, Amsterdam, 2002), pp. 885–982

    Google Scholar 

  11. J.C. Robinson, in Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2001)

    Google Scholar 

  12. S.G. Sell, Y. You, in Dynamics of Evolutionary Equations. Applied Mathematical Sciences, vol. 143 (Springer, Berlin, 2002)

    Google Scholar 

  13. R. Temam, in Dynamical Systems in Physics and Applications. Springer Texts in Applied Mathematics (Springer, Berlin, 1992)

    Google Scholar 

  14. T. Wakasa, Exact eigenvalues and eigenfunctions associated with linearization of Chafee-Infante equation. Funkcialaj Ekvacioj 49, 321–336 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

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Debussche, A., Högele, M., Imkeller, P. (2013). The Fine Dynamics of the Chafee–Infante Equation. In: The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise. Lecture Notes in Mathematics, vol 2085. Springer, Cham. https://doi.org/10.1007/978-3-319-00828-8_2

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