Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

This paper deals with the global existence and blow-up of solutions to a inhomogeneous pseudo-parabolic equation with initial value \begin{document}$ u_0 $\end{document} in the Sobolev space \begin{document}$ H_0^1( \Omega) $\end{document} , where \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} ( \begin{document}$ n\geq1 $\end{document} is an integer) is a bounded domain. By using the mountain-pass level \begin{document}$ d $\end{document} (see (14)), the energy functional \begin{document}$ J $\end{document} (see (12)) and Nehari function \begin{document}$ I $\end{document} (see (13)), we decompose the space \begin{document}$ H_0^1( \Omega) $\end{document} into five parts, and in each part, we show the solutions exist globally or blow up in finite time. Furthermore, we study the decay rates for the global solutions and lifespan (i.e., the upper bound of blow-up time) of the blow-up solutions. Moreover, we give a blow-up result which does not depend on \begin{document}$ d $\end{document} . By using this theorem, we prove the solution can blow up at arbitrary energy level, i.e. for any \begin{document}$ M\in \mathbb{R} $\end{document} , there exists \begin{document}$ u_0\in H_0^1( \Omega) $\end{document} satisfying \begin{document}$ J(u_0) = M $\end{document} such that the corresponding solution blows up in finite time.