In this paper, we consider the asymptotic behavior of the Boussinesq equation with nonlocal weak damping when the nonlinear function is arbitrary polynomial growth. We firstly prove the well-posedness of solution by means of the monotone operator theory. At the same time, we obtain the dissipative property of the dynamical system (E ,S( t)) associated with the problem in the space H 0 2 ( Ω ) × L 2 ( Ω ) and D ( A 3 4 ) × H 0 1 ( Ω ) , respectively. After that, the asymptotic smoothness of the dynamical system (E ,S( t)) and the further quasi-stability are demonstrated by the energy reconstruction method. Finally, different from [21] we show not only existence of the finite global attractor but also existence of the generalized exponential attractor.

In this paper, we consider the strong instability of standing waves for the Hartree equation with a partial/complete harmonic potential where . By establishing the corresponding profile decomposition theory, we first obtain the variational characterization of ground state solutions. Then, we deduce that there exists such that for all , the ground state standing wave is strongly unstable by blow‐up. Our results give an improvement for some recent results.

In this article, we consider blow-up criteria and instability of standing waves for the fractional Schrodinger-Poisson equation. By using the localized virial estimates, we establish the blow-up criteria for non-radial solutions in both mass-critical and mass-supercritical cases. Based on these blow-up criteria and three variational characterizations of the ground state, we prove that the standing waves are strongly unstable. These obtained results extend the corresponding ones presented in the literature.