Longtime dynamics of Boussinesq type equations with fractional damping

The paper investigates the well-posedness and longtime dynamics of Boussinesq type equations with fractional damping: utt+Δ2u+(−Δ)αut−Δf(u)=g(x), with α∈(1,2). The main results focus on the relations among the dissipative exponent α, the growth exponent p of nonlinearity f(u) and the well-posedness and the longtime dynamics of the equations. We find a new critical exponent pα≡N+2(2α−1)(N−2(2α−1))+ rather than p∗≡N+2N−2(N≥3) as known before and show that when 1≤p0. (ii) The related solution semigroup has a global attractor Aα in natural energy space, and also has an exponential attractor Aexpα in the sense of partially strong topology. In particular, when 1≤p