|
This article is cited in 19 scientific papers (total in 19 papers)
Approximation of trajectories lying on a global attractor of a hyperbolic equation with exterior force rapidly oscillating in time
M. I. Vishik, V. V. Chepyzhov Institute for Information Transmission Problems, Russian Academy of Sciences
Abstract:
A quasilinear dissipative wave equation is considered for periodic boundary conditions with exterior force $g(x,t/\varepsilon)$ rapidly oscillating in $t$. It is assumed in addition that, as $\varepsilon\to0+$, the function $g(x,t/\varepsilon)$ converges in the weak sense
(in $L_{2,w}^{\mathrm{loc}}(\mathbb R,L_2(\mathbb T^n))$ to a function
$\overline g(x)$ and the averaged wave equation (with exterior force $\overline g(x)$ has only finitely many stationary points $\{z_i(x),\,i= 1,\dots,N\}$, each of them hyperbolic. It is proved that the global attractor $\mathscr A_\varepsilon$ of the original equation deviates in the energy norm from the global attractor $\mathscr A_0$ of the averaged equation by a quantity $C\varepsilon^\rho$, where $\rho$ is described by an explicit formula. It is also shown that each piece of a trajectory $u^\varepsilon(t)$ of the original equation lying on $\mathscr A_\varepsilon$ that corresponds to an interval of time-length $C\log(1/\varepsilon)$
can be approximated to within $C_1\varepsilon^{\rho_1}$ by means of finitely many pieces of trajectories lying on unstable manifolds $M^u(z_i)$ of the averaged equation, where an explicit expression for $\rho_1$ is provided.
Received: 21.03.2003
Citation:
M. I. Vishik, V. V. Chepyzhov, “Approximation of trajectories lying on a global attractor of a hyperbolic equation with exterior force rapidly oscillating in time”, Mat. Sb., 194:9 (2003), 3–30; Sb. Math., 194:9 (2003), 1273–1300
Linking options:
https://www.mathnet.ru/eng/sm765https://doi.org/10.1070/SM2003v194n09ABEH000765 https://www.mathnet.ru/eng/sm/v194/i9/p3
|
Statistics & downloads: |
Abstract page: | 566 | Russian version PDF: | 217 | English version PDF: | 2 | References: | 71 | First page: | 3 |
|