Advances in Differential Equations

The regular attractor for the reaction-diffusion system with a nonlinearity rapidly oscillating in time and its averaging

M. Efendiev and S. Zelik

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Abstract

The longtime behaviour of solutions of a reaction-diffusion system with the nonlinearity rapidly oscillating in time ($f=f(t/{\varepsilon},u)$) is studied. It is proved that (under the natural assumptions) this behaviour can be described in terms of global (uniform) attractors $\mathcal A^{\varepsilon}$ of the corresponding dynamical process and that these attractors tend as ${\varepsilon}\to 0$ to the global attractor $\mathcal A^0$ of the averaged autonomous system. Moreover, we give a detailed description of the attractors $\mathcal A^{\varepsilon}$, ${\varepsilon}\ll1$, in the case where the averaged system possesses a global Liapunov function.

Article information

Source
Adv. Differential Equations Volume 8, Number 6 (2003), 673-732.

Dates
First available in Project Euclid: 19 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355926831

Mathematical Reviews number (MathSciNet)
MR1969651

Zentralblatt MATH identifier
1030.35016

Subjects
Primary: 35B41: Attractors
Secondary: 35K57: Reaction-diffusion equations 37L30: Attractors and their dimensions, Lyapunov exponents

Citation

Efendiev, M.; Zelik, S. The regular attractor for the reaction-diffusion system with a nonlinearity rapidly oscillating in time and its averaging. Adv. Differential Equations 8 (2003), no. 6, 673--732. https://projecteuclid.org/euclid.ade/1355926831.


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