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Application: The Chaffee—Infante Reaction—Diffusion Equation

  • P. Constantin
  • C. Foias
  • B. Nicolaenko
  • R. Teman
Part of the Applied Mathematical Sciences book series (AMS, volume 70)

Abstract

As an example of a parabolic reaction—diffusion equation with less stringent conditions than in Chapter 18, we briefly outline the construction of an inertial manifold for the Chaffee—Infante equation [H] in two dimensions:
$$\frac{{\partial u}}{{\partial t}} - \Delta u + \lambda \left( {{u^3} - u} \right) = 0,\;\lambda > {\text{ }}0,\Omega = {\left[ { - \pi , + \pi } \right]^2} = {T^2},{\text{ periodic boundary conditions, }}u\left( 0 \right) = {u_0}$$
(19.1)
(we do not restrict ourselves to odd periodic functions). For λ > 1, this equation admits multiple nonconstant steady states besides u = 0 and u = ± 1.

Keywords

Integral Manifold Inertial Manifold Nontrivial Zero Weak Maximum Principle Universal Attractor 
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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Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • P. Constantin
    • 1
  • C. Foias
    • 2
  • B. Nicolaenko
    • 3
  • R. Teman
    • 4
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsIndiana UniversityBloomingtonUSA
  3. 3.Center for Nonlinear StudiesLos Alamos National LaboratoryLos AlamosUSA
  4. 4.Department de MathematiquesUniversité de Paris-SudOrsayFrance

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