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We study the first initial-boundary-value problem for the three-dimensional non-autonomous Navier-Stokes-Voigt equations in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. The existence of a weak solution to the problem is proved by using the Faedo-Galerkin method. We then show the existence of a unique minimal finite-dimensional pull-back Dσ-attractor for the process associated with the problem, with respect to a large class of non-autonomous forcing terms. We also discuss relationships between the pull-back attractor, the uniform attractor and the global attractor.

In this paper we study the decay characterization in the space H K σ (R n ) of solutions to the viscous Camassa-Holm equations (VCHE) in the whole space R n (n = 2, 3, 4), namely,where m + 2p K, r * = r * (v 0 ) is the decay character of the initial datumWe also get the optimal lower bounds for decay rates of solutions to VCHE when −n/2 < r * 1. In particular, when v 0 ∈ H K σ (R n ) ∩ L 1 (R n ) has decay character r * (v 0 ) = 0, then we recover the previous results of Bjorland

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