论文标题
对于复杂系数的四阶schrödinger方程的局部适应性和有限的时间爆炸
Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient
论文作者
论文摘要
我们考虑四阶Schrödinger方程$$ i \ partial_tu+δ^2 u+μΔU+μΔ首先，我们证明了$ h^4 \ left（\ r^n \ right）$ in $ h^4 $ subclitical和关键案例中的本地适合度：$ h^4 $ clitage and：$α> 0 $，$（n-8）α\ leq8 $。然后，对于任何给定的紧凑型集合$ k \ subset \ mathbb {r}^n $，我们构造了$ h^4（\ r^n）$ solutions $ t> 0 $的$（-t，0）$定义，并在$ t = 0 $上完全在$ k $上爆炸。
We consider the fourth-order Schrödinger equation $$ 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 $t=0$.
