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The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation:utt+Δ2u+(-Δ)αut-Δf(u)=g(x), with α∈(0,1). For general bounded domain Ω⊂RN(N≥1), we show that there exists a critical exponent pα≡N+2(2α-1)(N-2)+ depending on the dissipative index α such that when the growth p of the nonlinearity f(u) is up to the range: 1≤pα, (i) the weak solutions of the equations are of additionally global smoothness when t>0; (i) the related dynamical system possesses a global attractor Aα and an exponential attractor Aαexp in natural energy space for each α∈(0,1), respectively; (ⅲ) the family of global attractors {Aα} is upper semicontinuous at each point α0∈(0,1], i.e., for any neighborhood U of Aα0,Aα⊂U when |α-α0|≪1. These results extend those for structural damping case: α∈[1,2) in [31,32]. [ABSTRACT FROM AUTHOR]

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]

This work is focused on the long-time behavior of solutions to the singularly perturbed damped Boussinesq equation in a 3D case [ABSTRACT FROM AUTHOR]

Abstract: The paper studies the longtime dynamics of the damped Boussinesq equation . First, the existence of global solutions to the initial boundary value problem of the equation is obtained provided that the growth exponent of , say , is either non-supercritical (subcritical and critical) or supercritical, especially, the stability of solutions is established when is non-supercritical. Second, the existence of a global attractor and an exponential attractor for the related solution semigroup are respectively established in the non-supercritical case. [Copyright &y& Elsevier]