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

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Abstract

We study semilinear elliptic equations Δu + cux=f(u,∇ u) and Δ2u + cux,= f(u, Δu, ∇2u) in infinite cylinders (x,-y)∈R×Ω⊂Rn+1 using methods from dynamical systems theory. We construct invariant manifolds, which contain the set of bounded solutions and then study a singular limit c→∞, 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.

Original languageEnglish (US)
Pages (from-to)469-547
Number of pages79
JournalJournal of Dynamics and Differential Equations
Volume8
Issue number4
DOIs
StatePublished - Jan 1 1996

Keywords

  • Inertial manifolds
  • Singular perturbation
  • Traveling waves

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