Approximation of trajectories lying on a global attractor of a hyperbolic equation with exterior force rapidly oscillating in time
- Institute for Information Transmission Problems Russian Academy of Sciences, Moscow (Russian Federation)
A quasilinear dissipative wave equation is considered for periodic boundary conditions with exterior force g(x,t/{epsilon}) rapidly oscillating in t. It is assumed in addition that, as {epsilon}{yields}0+, the function g(x,t/{epsilon}) converges in the weak sense (in L{sub 2,w}{sup loc}(R,L{sub 2}(T{sup n})) to a function g-bar(x) and the averaged wave equation (with exterior force g-bar(x) has only finitely many stationary points {l_brace}z{sub i}(x), i= 1,...,N{r_brace}, each of them hyperbolic. It is proved that the global attractor A{sub {epsilon}} of the original equation deviates in the energy norm from the global attractor A{sub 0} of the averaged equation by a quantity C{epsilon}{sup {rho}}, where {rho} is described by an explicit formula. It is also shown that each piece of a trajectory u{sup {epsilon}}(t) of the original equation lying on A{sub {epsilon}} that corresponds to an interval of time-length C log (1/{epsilon}) can be approximated to within C{sub 1}{epsilon}{sup {rho}{sub 1}} by means of finitely many pieces of trajectories lying on unstable manifolds M{sup u}(z{sub i}) of the averaged equation, where an explicit expression for {rho}{sub 1} is provided.
- OSTI ID:
- 21208353
- Journal Information:
- Sbornik. Mathematics, Vol. 194, Issue 9; Other Information: DOI: 10.1070/SM2003v194n09ABEH000765; Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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