Journal of Dynamics and Differential Equations

, Volume 7, Issue 2, pp 365–373

Determining nodes for the Kuramoto-Sivashinsky equation

  • Ciprian Foias
  • Igor Kukavica
Article

DOI: 10.1007/BF02219361

Cite this article as:
Foias, C. & Kukavica, I. J Dyn Diff Equat (1995) 7: 365. doi:10.1007/BF02219361

Abstract

We show that solutions of the 1D Kuramoto-Sivashinsky equation with periodic boundary conditions are asymptotically determined by their values at four points. That is, there existx1,x2,x3, andx4 in the (periodic) domainΩ such that if
$$\mathop {\lim }\limits_{t \to \infty } \left| {u_1 (x_j ,t) - u_2 (x_j ,t)} \right| = 0, j = 1,2,3,4$$
for two solutionsu1 andu2, then
$$\mathop {\lim }\limits_{t \to \infty } \left\| {u_1 ( \cdot ,t) - u_2 ( \cdot ,t)} \right\|_{L^2 (\Omega )} = 0$$

Key words

Kuramoto-Sivashinsky equationdetermining nodesglobal attractordissipative partial differential equations

AMS classification numbers

35K2235K3035K5535Q35

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Ciprian Foias
    • 1
  • Igor Kukavica
    • 2
  1. 1.Department of MathematicsIndiana UniversityBloomington
  2. 2.Department of MathematicsUniversity of ChicagoChicago