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 Mat. Sb., 2005, Volume 196, Number 6, Pages 17–42 (Mi msb1363)

Non-autonomous Ginzburg–Landau equation and its attractors

M. I. Vishik, V. V. Chepyzhov

Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: The behaviour as $t\to+\infty$ of solutions $\{u(x,t), t\geqslant0\}$ of the non-autonomous Ginzburg–Landau (G.–L.) equation is studied. The main attention is focused on the case when the dispersion coefficient $\beta(t)$ in this equation satisfies the inequality $|\beta(t)|>\sqrt3$ for $t\in L$, where $L$ is an unbounded subset of $\mathbb R_+$. In this case the uniqueness theorem for the G.–L. equation is not proved. The trajectory attractor $\mathfrak A$ for this equation is constructed.
If the coefficients and the exciting force are almost periodic (a.p.) in time and the uniqueness condition fails, then the trajectory attractor $\mathfrak A$ is proved to consist precisely of the solutions $\{u(x,t), t\geqslant0\}$ of the G.-L. equation that admit a bounded extension as solutions of this equation onto the entire time axis $\mathbb R$.
The behaviour as $t\to+\infty$ of solutions of a perturbed G.–L. equation with coefficients and the exciting force that are sums of a.p. functions and functions approaching zero in the weak sense as $t\to+\infty$ is also studied.

DOI: https://doi.org/10.4213/sm1363

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English version:
Sbornik: Mathematics, 2005, 196:6, 791–815

Bibliographic databases:

UDC: 517.956
MSC: 35Q55, 35B41, 35B40, 35K55

Citation: M. I. Vishik, V. V. Chepyzhov, “Non-autonomous Ginzburg–Landau equation and its attractors”, Mat. Sb., 196:6 (2005), 17–42; Sb. Math., 196:6 (2005), 791–815

Citation in format AMSBIB
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Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Chepyzhov V.V., Titi E.S., Vishik M.I., “On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier–Stokes system”, Discrete Contin. Dyn. Syst., 17:3 (2007), 481–500
2. Vishik M.I., Chepyzhov V.V., “The global attractor of the nonautonomous 2D Navier–Stokes system with singularly oscillating external force”, Doklady Mathematics, 75:2 (2007), 236–239
3. M. I. Vishik, V. V. Chepyzhov, “Trajectory attractors of reaction-diffusion systems with small diffusion”, Sb. Math., 200:4 (2009), 471–497
4. Chepyzhov V.V., Vishik M.I., “Trajectory attractor of a reaction-diffusion system with a series of zero diffusion coefficients”, Russ. J. Math. Phys., 16:2 (2009), 208–227
5. M. I. Vishik, V. V. Chepyzhov, “Trajectory attractor for a reaction-diffusion system with a small diffusion coefficient”, Dokl. Math., 79:2 (2009), 227–230
6. M. I. Vishik, V. V. Chepyzhov, “Trajectory attractor for a system of two reaction-diffusion equations with diffusion coefficient $\delta(t)\to0+$ as $t\to+\infty$”, Dokl. Math., 81:2 (2010), 196–200
7. Chepyzhov V.V., Vishik M.I., “Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time”, Discrete Contin. Dyn. Syst., 27:4 (2010), 1493–1509
8. Vishik M.I., Zelik S.V., Chepyzhov V.V., “Strong trajectory attractor for a dissipative reaction-diffusion system”, Dokl. Math., 82:3 (2010), 869–873
9. M. I. Vishik, V. V. Chepyzhov, “Trajectory attractors of equations of mathematical physics”, Russian Math. Surveys, 66:4 (2011), 637–731
10. Hongyu Cheng, Jianguo Si, “Quasi-periodic solutions for the quasi-periodically forced cubic complex Ginzburg-Landau equation on $\mathbb T^d$”, J. Math. Phys., 54:8 (2013), 082702, 27 pp.
11. Yangrong Li, Hongyong Cui, “Pullback attractor for nonautonomous Ginzburg-Landau equation with additive noise”, Abstr. Appl. Anal., 2014 (2014), 921750, 10 pp.
12. V. V. Chepyzhov, “Approximating the trajectory attractor of the 3D Navier-Stokes system using various $\alpha$-models of fluid dynamics”, Sb. Math., 207:4 (2016), 610–638
13. Chechkin G.A. Chepyzhov V.V. Pankratov L.S., “Homogenization of Trajectory Attractors of Ginzburg-Landau Equations With Randomly Oscillating Terms”, Discrete Contin. Dyn. Syst.-Ser. B, 23:3 (2018), 1133–1154
14. Bekmaganbetov K.A. Chechkin G.A. Chepyzhov V.V., “Strong Convergence of Trajectory Attractors For Reaction-Diffusion Systems With Random Rapidly Oscillating Terms”, Commun. Pure Appl. Anal, 19:5 (2020), 2419–2443
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