Blowup and scattering problems for the nonlinear Schrödinger equations

Abstract

We consider $L^2$-supercritical and $H^1$-subcritical focusing nonlinear[4] Schrödinger equations. We introduce a subset $\mathrm{PW}$ of $H^1\left({\mathbb{R}}^d\right)$ for $d\ge 1$, and investigate behavior of the solutions with initial data in this set. To this end, we divide $\mathrm{PW}$ into two disjoint components ${\mathrm{PW}}_{+}$ and ${\mathrm{PW}}_{-}$. Then, it turns out that any solution starting from a datum in ${\mathrm{PW}}_{+}$ behaves asymptotically free, and solution starting from a datum in ${\mathrm{PW}}_{-}$ blows up or grows up, from which we find that the ground state has two unstable directions. Our result is an extension of the one by Duyckaerts, Holmer, and Roudenko to the general powers and dimensions, and our argument mostly follows the idea of Kenig and Merle.