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

Abstract

<p style='text-indent:20px;'>We consider the fourth-order Schrödinger equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\partial_tu+\Delta^2 u+\mu\Delta u+\lambda|u|^\alpha u = 0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \alpha>0, \mu = \pm1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M2">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \lambda\in\mathbb{C} $\end{document}</tex-math></inline-formula>. Firstly, we prove local well-posedness in <inline-formula><tex-math id="M4">\begin{document}$ H^4\left( {\mathbb R}^N\right) $\end{document}</tex-math></inline-formula> in both <inline-formula><tex-math id="M5">\begin{document}$ H^4 $\end{document}</tex-math></inline-formula> subcritical and critical case: <inline-formula><tex-math id="M6">\begin{document}$ \alpha>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ (N-8)\alpha\leq8 $\end{document}</tex-math></inline-formula>. Then, for any given compact set <inline-formula><tex-math id="M8">\begin{document}$ K\subset\mathbb{R}^N $\end{document}</tex-math></inline-formula>, we construct <inline-formula><tex-math id="M9">\begin{document}$ H^4( {\mathbb R}^N) $\end{document}</tex-math></inline-formula> solutions that are defined on <inline-formula><tex-math id="M10">\begin{document}$ (-T, 0) $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M11">\begin{document}$ T>0 $\end{document}</tex-math></inline-formula>, and blow up exactly on <inline-formula><tex-math id="M12">\begin{document}$ K $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M13">\begin{document}$ t = 0 $\end{document}</tex-math></inline-formula>.</p>