Stability for a class of nonlinear pseudo-differential equations

✍ Scribed by Michael Frankel; Victor Roytburd

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 210 KB
• Volume: 21
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/j.aml.2007.03.023

✦ Synopsis

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(u x ) where cη 2 ≤ G(η) ≤ Cη 2 for 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.