Realization of vector fields and dynamics of spatially homogeneous parabolic equations

E. N. Dancer, P. Poláčik

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Abstract

We consider the Dirichlet problem for semilinear parabolic equations of the form ut = Δu + h(u, ∇u), t > 0, x ∈ Ω, (1) on a smooth bounded domain Ω ⊂ ℝN, N ≥ 2. The nonlinearity h : ℝ × ℝN → ℝ is assumed continuously difierentiable and spatially homogeneous (that is, independent of x). Applying the method of realization of vector fields, we show that (1) can generate very complicated dynamics. For example, choosing h and Ω suitably, one achieves that (1) has an invariant manifold W the flow on which has a transverse homoclinic orbit to a periodic orbit. Another choice of h and Ω yields a transitive Anosov flow on W. The invariant manifold W can have any dimension n ≥ 3 while N = dimΩ is fixed (greater then 1). In particular, the solutions of (1) can have ω-limit sets of arbitrarily high dimensions even though Ω has low dimension. We describe the method of realization of vector fields in detail, first in an abstract context and then for the above specific class of equations. A large piece of our analysis is devoted to the study of perturbations of multiple eigenvalues and corresponding eigenfunctions of the Laplacian under perturbation of the domain.

Original languageEnglish (US)
JournalMemoirs of the American Mathematical Society
Volume140
Issue number668
DOIs
StatePublished - Jul 1999

Keywords

  • Anosov flow
  • Elliptic eigenvalue problems
  • Invariant manifolds
  • Perturbation of the domain
  • Reaction-diffusion equations
  • Realization of vector fields
  • Spatially homogeneous parabolic equations
  • Transverse homoclinic orbit

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