Longtime dynamics of Boussinesq type equations with fractional damping

Abstract

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≤p<pα: (i) The equations are like parabolic, that is, not only the IBVP of the equations are well-posedness, but also their weak solutions are of higher global regularity as t>0. (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<pα′≡N+2α(N−2α)+(<pα), the partially strong topology becomes the strong one. (iii) For any α0∈[1,2), the family of global attractors Aα is upper semicontinuous at the point α0.