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Evolution equations and their trajectory attractors. (English) Zbl 0896.35032
Summary: We introduce the concept of a trajectory attractor for nonlinear evolution equations. Trajectory attractors seems to be an appropriate tool for the study of the long-time behavior of solutions of dynamical systems for which the corresponding initial-value Cauchy problem can have non-unique solution. An existence and perturbation theory is developed for a general non-autonomous evolution equations with translation-compact time symbols. By means of the trajectory attractors, we study 3D Navier-Stokes system with time-dependent external force, non-autonomous dissipative hyperbolic equation having an arbitrary polynomial growth of the nonlinear term, and other equations. For all these problems the uniqueness theorem of the Cauchy problem is not valid or it is not proved yet.

MSC:
35G25Initial value problems for nonlinear higher-order PDE
35B40Asymptotic behavior of solutions of PDE
35L70Nonlinear second-order hyperbolic equations
35Q30Stokes and Navier-Stokes equations