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

TL;DR

This paper studies the global behavior of solutions to a non-radial, energy-critical inhomogeneous biharmonic nonlinear Schrödinger equation, establishing conditions for global existence and scattering without symmetry assumptions.

Contribution

It extends the analysis of inhomogeneous NLS to the fourth-order case in a non-radial setting, developing new techniques to handle the lack of symmetry and conserved quantities.

Findings

Proves global well-posedness and scattering under energy bounds.

Develops a refined concentration-compactness and rigidity framework.

Addresses analytical challenges from inhomogeneity and higher-order dispersion.

Abstract

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.