论文标题
聚焦不均匀分数schrödinger方程的长时间动力学
Long time dynamics for the focusing inhomogeneous fractional Schrödinger equation
论文作者
论文摘要
我们考虑以下分数nls，将注意力集中到不均匀的功率型非线性$$ i \ partial_t u - （ - δ） $ n \ geq 2 $，$ 1/2 <s <1 $，$ 0 <b <2s $和$ 1+\ frac {2（2s-b）} {n} <p <1+ \ frac {2（2s-b）} {n-2s} $。 我们证明了全球存在和散射与有限的时间爆破临界状态中的能量解决方案的基态阈值，并具有球形对称的初始数据。 Dodson-Murphy的新方法证明了散射（{Proc。Am。Math。Soc。} {145}：{4859---4867}，2017年）。该方法基于陶的散射标准和摩拉维兹的估计。一个人在DINH最近的论文精神的精神（cont。Dyn。Syst。40：6441---6471，2020）的精神上描述了阈值。径向假设避免了Strichartz估计值的规律性丧失。这里的挑战是要克服两个主要困难。第一个是非本地分数拉普拉斯操作员的存在。第二个是非线性中的奇异重量。本文的大部分内容致力于证明$ h^s（\ mathbb {r}^n）$中全局解决方案的散射。
We consider the following fractional NLS with focusing inhomogeneous power-type nonlinearity $$i\partial_t u -(-Δ)^s u + |x|^{-b}|u|^{p-1}u=0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^N,$$ where $N\geq 2$, $1/2<s<1$, $0<b<2s$ and $1+\frac{2(2s-b)}{N}<p<1+\frac{2(2s-b)}{N-2s}$. We prove the ground state threshold of global existence and scattering versus finite time blow-up of energy solutions in the inter-critical regime with spherically symmetric initial data. The scattering is proved by the new approach of Dodson-Murphy ({Proc. Am. Math. Soc.} {145}: {4859--4867}, 2017). This method is based on Tao's scattering criteria and Morawetz estimates. One describes the threshold using some non-conserved quantities in the spirit of the recent paper by Dinh (Discr. Cont. Dyn. Syst. 40: 6441--6471, 2020). The radial assumption avoids a loss of regularity in Strichartz estimates. The challenge here is to overcome two main difficulties. The first one is the presence of the non-local fractional Laplacian operator. The second one is the presence of a singular weight in the non-linearity. The greater part of this paper is devoted to prove the scattering of global solutions in $H^s(\mathbb{R}^N)$.
