标题： 关于通过双轨道轮廓分解的质量-能量双重临界NLS的精确散射阈值 显示英文标题

标题： On sharp scattering threshold for the mass-energy double critical NLS via double track profile decomposition

摘要： 本文关注质量-能量双重临界NLS \begin{align} i\partial_t u+\Delta u\pm |u|^{\frac{4}{d}}u\pm |u|^{\frac{4}{d-2}}u=0\tag{DCNLS} \end{align} 在 $H^1(\mathbb{R}^d)$ 中的大数据散射问题，其中 $d\geq 3$。 在非聚焦-非聚焦情形下，Tao、Visan和Zhang证明了DCNLS的唯一解对于任意初始数据在 $H^1(\mathbb{R}^d)$ 中都是全局且时间散射的。 当至少一个非线性项为聚焦时，这不成立，因为可能形成爆破解和孤立子解。 然而，在所有剩余情形中，DCNLS解散射的精确阈值仍然是开放的。 遵循经典的集中紧致原理，我们在所有剩余情形中根据基态给出了DCNLS的精确散射阈值。 新的挑战来自于标准$L^2$- 或$\dot{H}^1$- 分解的余项无法同时具有渐近消失的对角线$L^2$- 和$\dot{H}^1$- Strichartz 范数。 为克服这一困难，我们构建了一个双轨道的剖面分解，该分解能够在单一剖面分解中捕捉低、中和高频泡，并且其余项在对角线$L^2$- 和$\dot{H}^1$- Strichartz 空间中都是渐近小的。 显示英文摘要

摘要： The present paper is concerned with the large data scattering problem for the mass-energy double critical NLS \begin{align} i\partial_t u+\Delta u\pm |u|^{\frac{4}{d}}u\pm |u|^{\frac{4}{d-2}}u=0\tag{DCNLS} \end{align} in $H^1(\mathbb{R}^d)$ with $d\geq 3$. In the defocusing-defocusing regime, Tao, Visan and Zhang show that the unique solution of DCNLS is global and scattering in time for arbitrary initial data in $H^1(\mathbb{R}^d)$. This does not hold when at least one of the nonlinearities is focusing, due to the possible formation of blow-up and soliton solutions. However, precise thresholds for a solution of DCNLS being scattering were open in all the remaining regimes. Following the classical concentration compactness principle, we impose sharp scattering thresholds in terms of ground states for DCNLS in all the remaining regimes. The new challenge arises from the fact that the remainders of the standard $L^2$- or $\dot{H}^1$-profile decomposition fail to have asymptotically vanishing diagonal $L^2$- and $\dot{H}^1$-Strichartz norms simultaneously. To overcome this difficulty, we construct a double track profile decomposition which is capable to capture the low, medium and high frequency bubbles within a single profile decomposition and possesses remainders that are asymptotically small in both of the diagonal $L^2$- and $\dot{H}^1$-Strichartz spaces.