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Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time. (English) Zbl 0668.35084

The Korteweg-de Vries equation was originally derived as a model for unidirectional waves of small amplitude, occurring in a large variety of physical situations in which nonlinearity and dispersion are important and have comparable effect. In many real problems, one cannot neglect energy dissipation mechanisms and external excitation. In this paper, the equation

u t +uu x +u xxx +γu=f

is considered to possess space-periodic solutions in the case where the external excitation f is either time-independent or time-periodic. The paper is organized as follows. The second section deals with the construction of the universal attractor for a nonlinear group, based on time-uniform estimates that follow from the use of the first three classical polynomial multiplies. The third section contains a result on the finite dimension of the universal attractor. Finally, an Appendix is devoted to an abstract result on the evolution of Gram determinants.

Reviewer: L.Y.Shih

MSC:
35Q99PDE of mathematical physics and other areas
35B40Asymptotic behavior of solutions of PDE
37C70Attractors and repellers, topological structure