Abstract:
We study a uniform attractor Aε for a dissipative wave equation in a bounded domain Ω⋐Rn under the assumption that the external force singularly oscillates in time; more precisely, it is of the form g0(x,t)+ε−αg1(x,t/ε), x∈Ω, t∈R, where α>0, 0<ε⩽1. 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.
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
This publication is cited in the following 12 articles:
Andrew Comech, Alexander Komech, Mikhail Vishik, Trends in Mathematics, Partial Differential Equations and Functional Analysis, 2023, 259
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
Xueli Song, Jianhua Wu, “Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor”, EECT, 11:1 (2022), 41
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
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
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
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
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
Yan X., “Dynamical Behaviour of Non-Autonomous 2D Navier–Stokes Equations with Singularly Oscillating External Force”, Dynam. Syst., 26:3 (2011), 245–260
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
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
Vladimir Chepyzhov, Mark Vishik, International Mathematical Series, 6, Instability in Models Connected with Fluid Flows I, 2008, 135