Attractors and their stability on Boussinesq type equations with gentle dissipation
The authors consider the Boussinesq type equations \[ \begin{cases} u_{tt}+\Delta^2u+(-\Delta)^\alpha u_t-\Delta f(u)=g(x) \quad \text{in}\;\Omega\times\mathbb{R}^+, \\ u_{\mid\partial\Omega}=\Delta u_{\mid\partial\Omega}=0, \quad u(x,0)=u_0(x),\;u_t(x,0)=u_1(x). \end{cases} \] Notably, \(\alpha\in(0,1)\) indicates the so-called gentile dissipation regime. Careful effort is given to clarifying the critical exponent of the nonlinearity in terms of \(\alpha\) and the spatial dimension \(N\), for \(\Omega\subset\mathbb{R}^N\). Indeed, the critical exponent is \(p_\alpha:=\frac{N+2(2\alpha-1)}{N-2}\), and the admissible polynomial growth exponents range \(1\leq p<p_\alpha\). Also under consideration is the asymptotic behavior in terms of global attractors and the continuity properties of these attractors with respect to the parameter \(\alpha.\) The interested reader can read the case for \(\alpha\in[1,2)\) in [the first author, J. Math. Anal. Appl. 399, No. 1, 180--190 (2013; Zbl 1256.35094)] and [the first and second author, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 161, 108--130 (2017; Zbl 1469.35193)].
scientific article; zbMATH DE number 6996073
scientific article