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Attractors and their stability on Boussinesq type equations with gentle dissipation

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The authors are supported by NNSF of China (No. 11671367)
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  • The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation:$ u_{tt}+Δ^2 u+(-Δ)^{α} u_{t}-Δ f(u) = g(x)$, with $α∈ (0, 1)$. For general bounded domain $Ω\subset \mathbb{R}^N (N≥1)$, we show that there exists a critical exponent $p_α\equiv\frac{N+2(2α-1)}{(N-2)^+}$ depending on the dissipative index α such that when the growth p of the nonlinearity f(u) is up to the range: $1≤p <p_α$, (ⅰ) the weak solutions of the equations are of additionally global smoothness when $t>0$; (ⅱ) the related dynamical system possesses a global attractor $\mathcal{A}_α$ and an exponential attractor $\mathcal{A}^α_{exp}$ in natural energy space for each $α∈ (0, 1)$, respectively; (ⅲ) the family of global attractors $\{\mathcal{A}_α\}$ is upper semicontinuous at each point $α_0∈ (0,1] $, i.e., for any neighborhood U of $\mathcal{A}_{α_0}, \mathcal{A}_α\subset U$ when $|α-α_0|\ll 1$. These results extend those for structural damping case: $α∈ [1, 2)$ in [31,32].

    Mathematics Subject Classification: Primary: 35B40, 35B41, 35B65; Secondary: 37L30, 37L15.


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