Scattering for the non-radial inhomogenous biharmonic NLS equation

Abstract

We consider the focusing inhomogeneous biharmonic nonlinear Schrödinger equation in $H^2({\mathbb{R}}^N)$, \begin{equation} iu_t + Δ^2 u - |x|^{-b}|u|^αu=0 \end{equation} when $b>0$ and $N\ge 5$. 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.