Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity

Abstract

<p style='text-indent:20px;'>To understand the characteristics of dynamical behavior especially the kinetic evolution for logarithmic nonlinearity, we aim to study a sixth-order Boussinesq equation with logarithmic nonlinearity in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ n\geq1 $\end{document}</tex-math></inline-formula> is an integer) with smooth boundary <inline-formula><tex-math id="M3">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>, where the dispersive and the strong damping are taken into account. Based on the Faedo-Galërkin method, the logarithmic Sobolev inequality, and the potential well method, the main ingredient of this paper is to construct several conditions for initial data leading to the solution global existence or infinite time blow-up, and to study the polynomial decay and the exponential decay of the energy of the system.