Scattering for the non-radial inhomogenous biharmonic NLS equation

TL;DR

This paper studies the scattering behavior of solutions to the focusing inhomogeneous biharmonic nonlinear Schrödinger equation in higher dimensions, establishing global results and scattering without radial symmetry assumptions.

Contribution

It proves scattering below the mass-energy threshold for non-radial initial data, extending previous results and simplifying the proof method.

Findings

Global small data results in $H^2$

Scattering below the threshold without radiality

Improved results in five dimensions

Abstract

We consider the focusing inhomogeneous biharmonic nonlinear Schr\"odinger equation in $H^2(\mathbb{R}^N)$, \begin{equation} iu_t + \Delta^2 u - |x|^{-b}|u|^{\alpha}u=0 \end{equation} when $b > 0$ and $N \geq 5$. We first obtain a small data global result in $H^2$, which, in the five-dimensional case, improves a previous result from Pastor and the second author. In the sequel, we show the main result, scattering below the mass-energy threshold in the intercritical case, that is, $\frac{8-2b}{N} < \alpha <\frac{8-2b}{N-4}$, without assuming radiality of the initial data. The proof combines the decay of the nonlinearity with Virial-Morawetz-type estimates to avoid the radial assumption, allowing for a much simpler proof than the Kenig-Merle roadmap.