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The center manifold technique and complex dynamics of parabolic equations

  • Krzysztof P. Rybakowski
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Part of the NATO ASI Series book series (ASIC, volume 472)

Abstract

Consider the following scalar parabolic equation
(1)
with Dirichlet boundary condition
(2)
or Neumann boundary condition
(3)
Here Ω ⊂ ℝ N is a smooth bounded domain (most frequently just the open unit ball), L is a second order selfadjoint uniformly elliptic differential operator with smooth coefficients (most often Lu = Δu + a(x)u where a is a smooth function on ) and v is the outer normal to the boundary ∂Ω of Ω. Finally,
$$ f:\left( {x,s,w} \right) \in \bar \Omega \times \mathbb{R} \times \mathbb{R}^N \mapsto f\left( {x,s,w} \right) \in \mathbb{R} $$
is some nonlinearity.

In these lectures we describe some recent realization results for vector fields or jets of vector fields on invariant manifolds of Eq. (1)–(2) and (1)–(3) and give an example of Eq. (1)–(2) with gradient independent nonlinearity and admitting a nonconvergent bounded trajectory. Some necessary mathematical background (center manifold theorem, Nash-Moser inverse mapping theorem) is also provided.

Keywords

Vector Field Parabolic Equation Dirichlet Boundary Condition Neumann Boundary Condition Invariant Manifold 
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 Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Krzysztof P. Rybakowski
    • 1
  1. 1.Dipartimento di Scienze MatematicheUniversità degli Studi di TriesteTriesteItaly

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