We study the well-posedness and longtime dynamics of the β-evolution equation with fractional damping: on the whole space , with β > 2α > 0. First, we find a critical exponent 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 in the energy space. The main difficulties come from the non-compactness of the Sobolev embedding on and the nonlocal characteristic of the equation. We overcome them by establishing some new commutator estimates.
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