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

Xuan Liu; Ting Zhang

doi:10.3934/dcdsb.2021156

Article Contents

2022, Volume 27, Issue 5: 2721-2757. Doi: 10.3934/dcdsb.2021156

This issue Previous Article The eigenvalue problem for a class of degenerate operators related to the normalized

$ p $

-Laplacian Next Article Corrigendum: On the Abel differential equations of third kind

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

Xuan Liu and
Ting Zhang^,

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

^* Corresponding author: Ting Zhang
^* Corresponding author: Ting Zhang

Received: November 30, 2020

Early access: June 2021

Published: May 2022

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Abstract

We consider the fourth-order Schrödinger 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( {\mathbb 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( {\mathbb R}^N) $ solutions that are defined on $ (-T, 0) $ for some $ T>0 $, and blow up exactly on $ K $ at $ t = 0 $.

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Mathematics Subject Classification: Primary: 35L71; Secondary: 35B30, 35B44.

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