The attractor for a nonlinear hyperbolic equation in the unbounded domain
By Sergey Zelik. 2001
74 citations recorded by the source corpus.
We study the long-time behavior of solutions for damped nonlinear hyperbolic equations in the unbounded domains. It is proved that under the natural assumptions these equations possess the locally compact attractors which may have the infinite Hausdorff and fractal dimension. That is why we obtain the upper and lower bounds for the Kolmogorov's entropy of these attractors. Moreover, we study the particular cases of these equations where the attractors occurred to be finite dimensional. For such particular cases we establish that the attractors consist of finite collections of finite dimensional unstable manifolds and every solution stabilizes to one of the finite number of equilibria points.
Subfields: Computer Networks and Communications, Control and Systems Engineering
View Paper: https://www.semanticscholar.org/paper/3d13f39e222fb29a7a1909a7ea6218a05d3c651f
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In Computer Networks and Communications
- Chunyou Sun
- Alain Miranville
- Meihua Yang
- Sergey Zelik
- Songsong Lu
- Stefanie Sonner
- Alexandre Nolasco de Carvalho
- Ciprian G. Gal
In Control and Systems Engineering
- Chunyou Sun
- Alain Miranville
- Sergey Zelik
- Meihua Yang
- Songsong Lu
- Ciprian G. Gal
- Hao Wu
- Maurizio Grasselli
Citations in the corpus
- Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity (2003, background): “8) may be considered as a natural generalization of this principle to the case of unbounded domains; see also [17, 32, 40]).”
- Chapter 3 Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains (2008, methodology): “, [235] for nonlinear damped wave equations, [88] for degenerate parabolic equations, and [10] for the Navier–Stokes equations.”
- Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity (2008, methodology): “, the solution in the sense of distribution given in [16, 34].”
- Attractors for nonautonomous 2D Navier–Stokes equations with less regular normal forces☆ (2006, background): “, [6,10,29]), especially to those systems without unique solvability.”