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: u t t + Δ 2 u + ( − Δ ) α u t − Δ f ( 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 p α ≡ N + 2 ( 2 α − 1 ) ( N − 2 ( 2 α − 1 ) ) + rather than p ∗ ≡ N + 2 N − 2 ( N ≥ 3 ) as known before and show that when 1 ≤ p < p α : (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 A α in natural energy space, and also has an exponential attractor A e x p α in the sense of partially strong topology. In particular, when 1 ≤ p < p α ′ ≡ N + 2 α ( N − 2 α ) + ( < p α ) , the partially strong topology becomes the strong one. (iii) For any α 0 ∈ [ 1 , 2 ) , the family of global attractors A α is upper semicontinuous at the point α 0 . [ABSTRACT FROM AUTHOR]