标题： 非径向能量临界非均匀双调和NLS的全局动力学 显示英文标题

标题： Global Dynamics of the Non-Radial Energy-Critical Inhomogeneous Biharmonic NLS

摘要： 我们研究能量临界情形下的聚焦非均匀四阶薛定谔方程\[ i\partial_t u + \Delta^2 u - |x|^{-b}|u|^p u = 0 \quad \text{on } \mathbb{R} \times \mathbb{R}^N, \]，其中$p = \frac{8 - 2b}{N - 4}$和$5 \leq N < 12$。 我们关注具有挑战性的非径向情形，并在次临界假设$ \sup_{t \in I} \|\Delta u(t)\|_{L^2} < \|\Delta W\|_{L^2}, $下建立全局适定性和散射，其中$W$表示相关椭圆方程的基态解。 与齐次情况下的先前结果（$b = 0$）不同，这些结果通常依赖于径向对称性和守恒量，我们的分析则不依赖对称性假设，并在非守恒量——动能下进行。 空间非均匀性与四阶色散算子的结合引入了重大的分析挑战。 为克服这些困难，我们开发了一个改进的集中紧致性和刚性框架，基于Kenig-Merle方法\cite{KM}，但更直接地受到Murphy和第一作者在二阶非齐次情形下的近期工作\cite{CM}的启发。 显示英文摘要

摘要： We investigate the focusing inhomogeneous nonlinear biharmonic Schr\"odinger equation \[ i\partial_t u + \Delta^2 u - |x|^{-b}|u|^p u = 0 \quad \text{on } \mathbb{R} \times \mathbb{R}^N, \] in the energy-critical regime, $p = \frac{8 - 2b}{N - 4}$, and $5 \leq N < 12$. We focus on the challenging non-radial setting and establish global well-posedness and scattering under the subcritical assumption $ \sup_{t \in I} \|\Delta u(t)\|_{L^2} < \|\Delta W\|_{L^2}, $ where $W$ denotes the ground state solution to the associated elliptic equation. In contrast to previous results in the homogeneous case ($b = 0$), which often rely on radial symmetry and conserved quantities, our analysis is carried out without symmetry assumptions and under a non-conserved quantity, the kinetic energy. The presence of spatial inhomogeneity combined with the fourth-order dispersive operator introduces substantial analytical challenges. To overcome these difficulties, we develop a refined concentration-compactness and rigidity framework, based on the Kenig-Merle approach \cite{KM}, but more directly inspired by recent work of Murphy and the first author \cite{CM} in the second-order inhomogeneous setting.