Abstract:
Sufficient conditions are obtained for the existence of a k-dimensional invariant manifold that attracts as t→∞ all solutions u(t) of the evolution equation ˙u=−Au+F(u) in a Hilbert space, where A is a linear selfadjoint operator, semibounded from below, with compact resolvent, and F is a uniformly Lipschitz (in suitable norms) nonlinearity; these conditions sharpen previously known conditions and cannot be improved.
Citation:
A. V. Romanov, “Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations”, Russian Acad. Sci. Izv. Math., 43:1 (1994), 31–47
\Bibitem{Rom93}
\by A.~V.~Romanov
\paper Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations
\jour Russian Acad. Sci. Izv. Math.
\yr 1994
\vol 43
\issue 1
\pages 31--47
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Linking options:
https://www.mathnet.ru/eng/im855
https://doi.org/10.1070/IM1994v043n01ABEH001557
https://www.mathnet.ru/eng/im/v57/i4/p36
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A. V. Romanov, “A Parabolic Equation with Nonlocal Diffusion without a Smooth Inertial Manifold”, Math. Notes, 96:4 (2014), 548–555
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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Yu. A. Goritsky, “Explicit construction of attracting integral manifolds for a dissipative hyperbolic equation”, J. Math. Sci. (N. Y.), 143:4 (2007), 3239–3252
A. Yu. Goritskii, V. V. Chepyzhov, “Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds”, Sb. Math., 196:4 (2005), 485–511
A. V. Romanov, “Finite-dimensional dynamics on attractors of non-linear parabolic equations”, Izv. Math., 65:5 (2001), 977–1001
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A. V. Romanov, “Three counterexamples in the theory of inertial manifolds”, Math. Notes, 68:3 (2000), 378–385
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