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Non-autonomous evolutionary equations with translation-compact symbols and their attractors. (English. Abridged French version) Zbl 0837.35059
Summary: We study a new class of non-autonomous evolutionary equations arising in mathematical physics. These equations contain time-dependent nonlinear functions and right-hand sides (i.e. time symbols of equations) which satisfy the translation-compact (tr.c.) property. The latter means that the set of all time translations of such functions from a precompact set with respect to the corresponding functional space topology. The class of symbols under consideration is much wider than the class of almost periodic (a.p.) symbol. We establish the existence and describe the structure of attractors for the equations with tr.c. symbols. For example, for 2D Navier-Stokes system with an external force \(\varphi(x, t)\), such that \(\int^{t+ 1}_t |\varphi(\cdot, s)|^2_H ds\leq M< +\infty\), \(\forall t\in \mathbb{R}\), we prove the existence of the uniform attractor and describe its structure. Similar results are established for the wide class of non-autonomous reaction-diffusion systems and other equations.

35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
58D25 Equations in function spaces; evolution equations
35B40 Asymptotic behavior of solutions to PDEs
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