Stability for a class of nonlinear pseudo-differential equations

  • Frankel M
  • Roytburd V
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Abstract

We study a class of nonlinear evolutionary equations generated by a pseudo-differential operator with the elliptic principal symbol and with nonlinearities of the form G (ux) where c η2≤ G (η) ≤ C η2for large | η |. We demonstrate existence of a universal absorbing set, and a compact attractor, and show that the attractor is of a finite Hausdorff dimension. The stabilization mechanism is similar to the nonlinear saturation well known for the Kuramoto-Sivashinsky equation. © 2007 Elsevier Ltd. All rights reserved.

Author-supplied keywords

  • Attractors
  • Dissipativity
  • Kuramoto-Sivashinsky equation
  • Pseudo-differential operators

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Authors

  • Michael Frankel

  • Victor Roytburd

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