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Matematicheskie Zametki, 2006, Volume 79, Issue 4, Pages 522–545
DOI: https://doi.org/10.4213/mzm2722
(Mi mzm2722)
 

This article is cited in 12 scientific papers (total in 12 papers)

Attractors of dissipative hyperbolic equations with singularly oscillating external forces

M. I. Vishik, V. V. Chepyzhov

Institute for Information Transmission Problems, Russian Academy of Sciences
References:
Abstract: We study a uniform attractor Aε for a dissipative wave equation in a bounded domain ΩRn under the assumption that the external force singularly oscillates in time; more precisely, it is of the form g0(x,t)+εαg1(x,t/ε), xΩ, tR, where α>0, 0<ε1. In E=H10×L2, this equation has an absorbing set Bε estimated as and, therefore, can increase without bound in the norm of E as \varepsilon\to0+. Under certain additional constraints on the function g_1(x,z), x\in\Omega, z\in\mathbb R, we prove that, for 0<\alpha\leqslant\alpha_0, the global attractors \mathscr A^\varepsilon of such an equation are bounded in E, i.e., \|\mathscr A^\varepsilon\|_E\leqslant C_3, 0<\varepsilon\leqslant1.
Along with the original equation, we consider a “limiting” wave equation with external force g_0(x,t) that also has a global attractor \mathscr A^0. For the case in which g_0(x,t)=g_0(x) and the global attractor \mathscr A^0 of the limiting equation is exponential, it is established that, for 0<\alpha\leqslant\alpha_0, the Hausdorff distance satisfies the estimate \operatorname{dist}_E(\mathscr A^\varepsilon,\mathscr A^0)\leqslant C\varepsilon^{\eta(\alpha)}, where \eta(\alpha)>0. For \eta(\alpha) and \alpha_0, explicit formulas are given. We also study the nonautonomous case in which g_0=g_0(x,t). It is assumed that sufficient conditions are satisfied for which the “limiting” nonautonomous equation has an exponential global attractor. In this case, we obtain upper bounds for the Hausdorff distance of the attractors \mathscr A^\varepsilon from \mathscr A^0, similar to those given above.
Received: 31.03.2005
English version:
Mathematical Notes, 2006, Volume 79, Issue 4, Pages 483–504
DOI: https://doi.org/10.1007/s11006-006-0054-2
Bibliographic databases:
UDC: 517.95
Language: Russian
Citation: M. I. Vishik, V. V. Chepyzhov, “Attractors of dissipative hyperbolic equations with singularly oscillating external forces”, Mat. Zametki, 79:4 (2006), 522–545; Math. Notes, 79:4 (2006), 483–504
Citation in format AMSBIB
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Linking options:
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  • https://doi.org/10.4213/mzm2722
  • https://www.mathnet.ru/eng/mzm/v79/i4/p522
  • This publication is cited in the following 12 articles:
    1. Andrew Comech, Alexander Komech, Mikhail Vishik, Trends in Mathematics, Partial Differential Equations and Functional Analysis, 2023, 259  crossref
    2. K. A. Bekmaganbetov, V. V. Chepyzhov, G. A. Chechkin, “Strong convergence of attractors of reaction-diffusion system with rapidly oscillating terms in an orthotropic porous medium”, Izv. Math., 86:6 (2022), 1072–1101  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. Xueli Song, Jianhua Wu, “Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor”, EECT, 11:1 (2022), 41  crossref
    4. 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  crossref  mathscinet  isi
    5. Bekmaganbetov K.A., Chechkin G.A., Chepyzhov V.V., ““Strange Term” in Homogenization of Attractors of Reaction-Diffusion Equation in Perforated Domain”, Chaos Solitons Fractals, 140 (2020), 110208  crossref  mathscinet  isi
    6. Bekmaganbetov K.A., Chechkin G.A., Chepyzhov V.V., “Weak Convergence of Attractors of Reaction-Diffusion Systems With Randomly Oscillating Coefficients”, Appl. Anal., 98:1-2, SI (2019), 256–271  crossref  mathscinet  isi  scopus
    7. Bekmaganbetov K.A., Chechkin G.A., Chepyzhov V.V., Goritsky A.Yu., “Homogenization of trajectory attractors of 3D Navier–Stokes system with randomly oscillating force”, Discret. Contin. Dyn. Syst., 37:5 (2017), 2375–2393  crossref  mathscinet  zmath  isi  scopus
    8. Chepyzhov V.V., Conti M., Pata V., “Averaging of Equations of Viscoelasticity With Singularly Oscillating External Forces”, J. Math. Pures Appl., 108:6 (2017), 841–868  crossref  mathscinet  zmath  isi  scopus
    9. Yan X., “Dynamical Behaviour of Non-Autonomous 2D Navier–Stokes Equations with Singularly Oscillating External Force”, Dynam. Syst., 26:3 (2011), 245–260  crossref  mathscinet  zmath  isi  elib  scopus
    10. Chepyzhov V. V., Pata V., Vishik M. I., “Averaging of nonautonomous damped wave equations with singularly oscillating external forces”, J. Math. Pures Appl. (9), 90:5 (2008), 469–491  crossref  mathscinet  zmath  isi  scopus
    11. Vishik M. I., Pata V., Chepyzhov V. V., “Time averaging of global attractors of nonautonomous wave equations with singularly oscillating external forces”, Dokl. Math., 78:2 (2008), 689–692  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    12. Vladimir Chepyzhov, Mark Vishik, International Mathematical Series, 6, Instability in Models Connected with Fluid Flows I, 2008, 135  crossref
    Citing articles in Google Scholar: Russian citations, English citations
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