Local well-posedness and finite time blowup for fourth-order Schr\"odinger equation with complex coefficient

TL;DR

This paper establishes local well-posedness for a complex coefficient fourth-order Schrödinger equation in certain Sobolev spaces and constructs solutions that blow up precisely on a given compact set at finite time.

Contribution

It proves local well-posedness in $H^4$ for the equation and constructs solutions with prescribed blow-up sets, extending understanding of blow-up phenomena for complex coefficient higher-order Schrödinger equations.

Findings

Proved local well-posedness in $H^4$ for subcritical and critical cases.

Constructed solutions that blow up exactly on any given compact set.

Demonstrated finite-time blow-up behavior with precise blow-up set control.

Abstract

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