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 \[ u_ t+uu_ x+u_{xxx}+\gamma u=f \] 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


35Q99 Partial differential equations of mathematical physics and other areas of application
35B40 Asymptotic behavior of solutions to PDEs
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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