Guo, Boling; Wu, Yonghui Global attractor and its dimension estimates for the generalized dissipative KdV equation on \(\mathbb{R}\). (English) Zbl 0932.37064 Acta Math. Appl. Sin., Engl. Ser. 14, No. 3, 252-259 (1998). The paper deals with a long-time behaviour generalized dissipative KdV equation in an unbounded domain, more precisely, in \(\mathbb{R}\). Due to the lack of compactness of \(H^2(\mathbb{R})\subset L^2(\mathbb{R})\), the authors use weighted Sobolev spaces as a candidate of phase space. They prove the asymptotical compactness of the corresponding semigroup and the existence of the global attractor. Based on Constantin-Foiaş-Temams formula the authors give an upper bound for the Hausdorff and fractal dimensions of the global attractor. Reviewer: Messoud Efendiev (Berlin) Cited in 1 Document MSC: 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37L50 Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems Keywords:generalized KdV equation; asymptotical compactness and smoothness; Hausdorff and fractal dimension; global attractor PDFBibTeX XMLCite \textit{B. Guo} and \textit{Y. Wu}, Acta Math. Appl. Sin., Engl. Ser. 14, No. 3, 252--259 (1998; Zbl 0932.37064) Full Text: DOI References: [1] F. Abergel. Existence and Finite Dimensionality of the Global Attractor for Evolution Equations on Unbounded Domain.J.D.E., 1990, 83(1): 85–108. · Zbl 0706.35058 [2] A.V. Babin. Asymptotics of Functions on Attractor for Two-dimensional Navier-Stokes System in Unbounded Plane Domain Under |x|.Math. Sb., 1991, 182(12): 1683–1709. · Zbl 0753.35064 [3] Ding Xiaqi and Wu Yonghui. Global Attractor and Its Finiteness of Two-dimensional Navier-Stokes Equations in Strip-like Domain.Acta Math. Sci., 1996, 16(2): 204–215. · Zbl 0909.35106 [4] J.M. Ghidaglia. Weakly Damped Forced KdV Equations Behave as a Finite Dimensional Dynamical System in the Long Time.J.D.E., 1988, 74: 369–390. · Zbl 0668.35084 [5] J. Ginibre and Y. Tsutsumi. Uniqueness of the Solutions for the Generalized KdV Equation.SIAM J. Math. Anal., 1989, 20(6): 1388–1425. · Zbl 0702.35224 [6] Guo Boling. The Viscosity Elimination Methods and the Viscosity of Difference Schemes. Chinese Academic Press, 1993. · Zbl 0777.65069 [7] Guo Boling and Yang Linge. The Global Attractors of the Periodic Initial Value Problem for a Coupled Nonlinear Wave Equations.Math. Method appl. Sci., 1996, 19: 131–144. · Zbl 0844.35106 [8] G.E. Kenig, G. Ponce and L. Vega. On the Generalized KdV Equations.Duke Math. J., 1989, 59(3): 581–610. · Zbl 0795.35105 [9] G.R. Sell and Y.C. You. Inertial Manifolds: the Non-self-adjoint Case.J.D.E., 1992, 96: 203–255. · Zbl 0760.34051 [10] R. Temam. Infinite Dimensional Dynamical Systems in Mech. and Phys. Springer-Verlag, 1988. · Zbl 0662.35001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.