Abstract. In this paper we develop and use successive averaging methods for explaining the regularization mechanism in the the periodic Korteweg-de Vries (KdV) equation in the homogeneous Sobolev spacesḢ s , for s ≥ 0. Specifically, we prove the global existence, uniqueness, and Lipschitz continuous dependence on the initial data of the solutions of the periodic KdV. For the case where the initial data is in L 2 we also show the Lipschitz continuous dependence of these solutions with respect to the initial data as maps fromḢ s toḢ s , for s ∈ (−1, 0].

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SynopsisThere is a large number of papers in which attractors of parabolic reaction-diffusion equations in bounded domains are investigated. In this paper, these equations are considered in the whole unbounded space, and a theory of attractors of such equations is built. While investigating these equations, specific difficulties arise connected with the noncompactness of operators, with the continuity of their spectra, etc. Therefore some new conditions on nonlinear terms arise. In this paper weighted spaces are widely applied. An important feature of this problem is worth mentioning: namely, properties of semigroups corresponding to equations with solutions in spaces of growing and of decreasing functions essentially differ.

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Estimates of the Hausdorff dimension of an attractor of a two-/KivioncirmQl Navipr-Stnkfis svstem 183 parabolic equations and parabolic systems 194 §10. Attractors of semigroups having a global Lyapunov function 202 §11. Regular attractors of semigroups having a Lyapunov function 206 References 210

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