Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient

We consider the fourth-order Schrödinger equation\begin{document}$ i\partial_tu+\Delta^2 u+\mu\Delta u+\lambda|u|^\alpha u = 0, $\end{document}where \begin{document}$ \alpha>0, \mu = \pm1 $\end{document} or \begin{document}$ 0 $\end{document} and \begin{document}$ \lambda\in\mathbb{C} $\end{document} . Firstly, we prove local well-posedness in \begin{document}$ H^4\left( {\mathbb R}^N\right) $\end{document} in both \begin{document}$ H^4 $\end{document} subcritical and critical case: \begin{document}$ \alpha>0 $\end{document} , \begin{document}$ (N-8)\alpha\leq8 $\end{document} . Then, for any given compact set \begin{document}$ K\subset\mathbb{R}^N $\end{document} , we construct \begin{document}$ H^4( {\mathbb R}^N) $\end{document} solutions that are defined on \begin{document}$ (-T, 0) $\end{document} for some \begin{document}$ T>0 $\end{document} , and blow up exactly on \begin{document}$ K $\end{document} at \begin{document}$ t = 0 $\end{document} .