A Virial-Morawetz approach to scattering for the non-radial inhomogeneous NLS

Abstract

Consider the focusing inhomogeneous nonlinear Schrödinger equation in $H^1({\mathbb{R}}^N)$, $$i{u}_t+\Delta u+|x{|}^{-b}|u{|}^{p-1}u=0,$$ when $b>0$ and $N\ge 3$ in the intercritical case $0<s_c<1$. In previous works, the second author, as well as Farah, Guzmán and Murphy, applied the concentration-compactness approach to prove scattering below the mass-energy threshold for radial and non-radial data. Recently, the first author adapted the Dodson-Murphy approach for radial data, followed by Murphy, who proved scattering for non-radial solutions in the 3d cubic case, for $b<1/2$. This work generalizes the recent result of Murphy, allowing a broader range of values for the parameters $p$ and $b$, as well as allowing any dimension $N\ge 3$. It also gives a simpler proof for scattering nonradial, avoiding the Kenig-Merle road map. We exploit the decay of the nonlinearity, which, together with Virial-Morawetz-type estimates, allows us to drop the radial assumption.

arXiv admin note: substantial text overlap with arXiv:1905.02663