We study the well-posedness and longtime dynamics of the β-evolution equation with fractional damping: ∂t2u+(−Δ)βu+γ(1−Δ)α∂tu+f(u)=g(x) on the whole space Rn, with β > 2α > 0. First, we find a critical exponent p*=n+4αn−2β for the well-posedness of energy solutions. In fact, if the nonlinear term grows with the order p ∈ [1, p*) and satisfies some dissipative conditions, then the equation is globally well-posed in the energy space. Moreover, both u and ∂tu have a smoothing effect as a parabolic equation. Finally, we show that the solution semigroup has a global attractor A in the energy space. The main difficulties come from the non-compactness of the Sobolev embedding on Rn and the nonlocal characteristic of the equation. We overcome them by establishing some new commutator estimates.

In this paper, we study the existence of pullback attractors and pullback exponential attractors for lattice dynamical system in time-dependent sequence space. First, we introduce a new sequence space with time-dependent variable exponents. Second, two abstract criteria (or sufficient conditions) about the existence of pullback attractors and pullback exponential attractors are established for infinite dimensional lattice dynamical systems on time-dependent spaces of infinite sequences. Finally, for making full use of the above-mentioned abstract criteria, we consider a second order lattice system with nonstandard growth nonlinearity, and then the existence of bi-space pullback attractors and pullback exponential attractors on a time-dependent Musielak–Orlicz space is obtained. In particular, we point out that these criteria and analytical skills can be utilized to deal with other lattice systems satisfying nonstandard growth conditions.