Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.02796 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911091736444928 |
|---|---|
| author | Guzmán, Carlos M. Keraani, Sahbi Xu, Chengbin |
| author_facet | Guzmán, Carlos M. Keraani, Sahbi Xu, Chengbin |
| contents | We investigate the focusing inhomogeneous nonlinear biharmonic Schrödinger equation \[ i\partial_t u + Δ^2 u - |x|^{-b}|u|^p u = 0 \quad \text{on } \mathbb{R} \times \mathbb{R}^N, \] 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} \|Δu(t)\|_{L^2} < \|ΔW\|_{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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02796 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Dynamics of the Non-Radial Energy-Critical Inhomogeneous Biharmonic NLS Guzmán, Carlos M. Keraani, Sahbi Xu, Chengbin Analysis of PDEs We investigate the focusing inhomogeneous nonlinear biharmonic Schrödinger equation \[ i\partial_t u + Δ^2 u - |x|^{-b}|u|^p u = 0 \quad \text{on } \mathbb{R} \times \mathbb{R}^N, \] 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} \|Δu(t)\|_{L^2} < \|ΔW\|_{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. |
| title | Global Dynamics of the Non-Radial Energy-Critical Inhomogeneous Biharmonic NLS |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.02796 |