| Abstract: |
| The paper investigates the well-posedness and longtime dynamics of Boussinesq type equations with fractional damping: $u_{tt}+\Delta^2 u+(-\Delta)^{\alpha} u_{t}-\Delta f(u)=g(x)$, with $\alpha\in (0,2)$. The main results focus on the relations among the dissipative exponent $\alpha$, 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 $p_\alpha$ depending on the dissipative index $\alpha$ such that when the growth $p$ of the nonlinearity $f(u)$ is up to the range $1\leq p0$. (ii) The related solution semigroup has a global attractor $\mathcal{A}_\alpha$ and an exponential attractor $\mathcal{A}^\alpha_{exp}$. (iii) For any $\alpha_0\in (0, 2)$, the family of global attractors $\mathcal{A}_\alpha$ is upper semicontinuous at the point $\alpha_0$. |
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