Abstract:
This paper is a study of an abstract model of a second-order non-linear elliptic boundary-value problem in a cylindrical domain by the methods of the theory of dynamical systems. It is shown that, under some natural conditions, the essential solutions of the problem in question are described by means of the global attractor of the corresponding trajectory dynamical system, and this attractor can have infinite fractal dimension and infinite topological entropy. Moreover, sharp upper and lower bounds are obtained for the Kolmogorov ε-entropy of these attractors.
Citation:
A. Mielke, S. V. Zelik, “Infinite-dimensional trajectory attractors of elliptic boundary-value problems in cylindrical domains”, Russian Math. Surveys, 57:4 (2002), 753–784
This publication is cited in the following 22 articles:
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Qin Yu., Wang X., “Upper Semicontinuity of Trajectory Attractors For 3D Incompressible Navier-Stokes Equation”, Appl. Math. Optim., 84:1 (2021), 1–18
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Mark Vishik, Sergey Zelik, “Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit”, CPAA, 13:5 (2014), 2059
Zelik S., “Inertial Manifolds and Finite-Dimensional Reduction For Dissipative PDEs”, Proc. R. Soc. Edinb. Sect. A-Math., 144:6 (2014), 1245–1327
A. Eden, S. V. Zelik, V. K. Kalantarov, “Counterexamples to regularity of Mañé projections in the theory of attractors”, Russian Math. Surveys, 68:2 (2013), 199–226
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Derks, G, “Perturbations of embedded eigenvalues for the bilaplacian on a cylinder”, Discrete and Continuous Dynamical Systems, 21:3 (2008), 801
A. Miranville, S. Zelik, Handbook of Differential Equations: Evolutionary Equations, 4, 2008, 103
Efendiev M., “On an elliptic attractor in an asymptotically symmetric unbounded domain in R4+”, Bull. Lond. Math. Soc., 39 (2007), 911–920
Mielke A., Zelik S.V., “Infinite-dimensional hyperbolic sets and spatio-temporal chaos in reaction diffusion systems in Rn”, J. Dynam. Differential Equations, 19:2 (2007), 333–389
A.V. Babin, Handbook of Dynamical Systems, 1, 2006, 983
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