Homogenization of attractors of non-linear hyperbolic equations with asymptotically degenerate coefficients

Journal Article · Sun Oct 31 04:00:00 EST 1999 · Sbornik. Mathematics

DOI:https://doi.org/10.1070/SM1999V190N09ABEH000427; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)· OSTI ID:21202884

Pankratov, L S ^[1]; Chueshov, I D ^[2]

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1. B.Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar'kov (Ukraine)
2. V.N. Karazin Kharkiv National University, Kharkiv (Ukraine)

A non-linear initial-boundary-value problem for a hyperbolic equation with dissipation is considered in a bounded domain {omega} u{sub tt}{sup {epsilon}} + {delta}u{sub t}{sup {epsilon}} - div(a{sup {epsilon}}(x){nabla}u{sup {epsilon}}) + f(u{sup {epsilon}}) = h{sup {epsilon}}(x) where {delta}>0 and the coefficient a{sup {epsilon}}(x) is of order {epsilon}{sup 3+{gamma}} (0{<=}{gamma}<1) on the union of spherical annuli of thickness d{sub {epsilon}}=d{epsilon}{sup 2+{gamma}}. The annuli are periodically, with period {epsilon}, distributed in a bounded domain {omega}. Outside the union of the annuli a{sup {epsilon}}(x){identical_to}1. The asymptotic behaviour of the solutions and the global attractor of the problem are studied as {epsilon}{yields}0. It is shown that the homogenization of the problem on each finite time interval leads to a system consisting of a non-linear hyperbolic equation and an ordinary second-order differential equation (with respect to t). It is also shown that the global attractor of the initial problem approaches in a certain sense a weak global attractor of the homogenized problem.

• OSTI ID: 21202884
• Journal Information: Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 9 Vol. 190; ISSN 1064-5616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ASYMPTOTIC SOLUTIONS
ATTRACTORS
BOUNDARY-VALUE PROBLEMS
DIFFERENTIAL EQUATIONS
NONLINEAR PROBLEMS
PERIODICITY
SPHERICAL CONFIGURATION