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Global averaging and parametric resonances in damped semilinear wave equations

Published online by Cambridge University Press:  12 July 2007

S. Zelik
S. Zelik
Affiliation:
Institut für Analysis, Dynamik und Modelierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
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Abstract

The long-time behaviour of solutions to a semilinear damped wave equation in a three-dimensional bounded domain with the nonlinearity rapidly oscillating in time (f = f(ε, u, t/ε)) is studied. It is proved that (under natural assumptions) the behaviour of solutions whose initial energy is not very large can be described in terms of global (uniform) attractors Aε of the corresponding dynamical processes and that, as ε → 0, these attractors tend to the global attractor A0 of the corresponding averaged system. We also give the detailed description of these attractors in the case where the limit attractor A0 is regular.

Moreover, we give explicit examples of semilinear hyperbolic equations where the uniform attractor Âε (for the initial data belonging to the whole energy phase space) contains the irregular resonant part, which tends to infinity as ε → 0, and formulate the additional restrictions on the nonlinearity f which guarantee that this part is absent.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 136 , Issue 5 , October 2006 , pp. 1053 - 1097
Copyright
Copyright © Royal Society of Edinburgh 2006

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