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arXiv · 2010.11055

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

Xuan Liu·Ting Zhang
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Abstract

We consider the fourth-order Schrödinger equation $$ i\partial_tu+Δ^2 u+μΔu+λ|u|^αu=0, $$ where $α>0,μ=\pm1$ or $0$ and $λ\in\mathbb{C}$. Firstly, we prove local well-posedness in $H^4\left(\R^N\right)$ in both $H^4$ subcritical and critical case: $α>0$, $(N-8)α\leq8$. 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 $t=0$.

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Xuan Liu, Ting Zhang. 2021-01-30. Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient. https://arxiv.org/abs/2010.11055

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