The nonlinear Schrödinger equation with combined power-type nonlinearities

Abstract

We undertake a comprehensive study of the nonlinear Schrödinger equation $$i{u}_t+\Delta u={\lambda }_1|u{|}^{p_1}u+{\lambda }_2|u{|}^{p_2}u,$$ where $u(t,x)$ is a complex-valued function in spacetime ${\mathbb{R}}_t\times {\mathbb{R}}_x^n$, ${\lambda }_1$ and ${\lambda }_2$ are nonzero real constants, and $0<p_1<p_2\le \frac{4}{n-2}$. We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space $H^1({\mathbb{R}}^n)$ and in the pseudoconformal space $\Sigma :=\{f\in H^1({\mathbb{R}}^n);xf\in L^2({\mathbb{R}}^n)\}$. Of particular interest is the case when both nonlinearities are defocusing and correspond to the $L_x^2$-critical, respectively ${\dot{H}}_x^1$-critical NLS, that is, ${\lambda }_1,{\lambda }_2>0$ and $p_1=\frac{4}{n}$, $p_2=\frac{4}{n-2}$. The results at the endpoint $p_1=\frac{4}{n}$ are conditional on a conjectured global existence and spacetime estimate for the $L_x^2$-critical nonlinear Schrödinger equation. As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in $H_x^1$ for solutions to the nonlinear Schrödinger equation $$i{u}_t+\Delta u=|u{|}^{p}u,$$ with $\frac{4}{n}<p<\frac{4}{n-2}$, which was first obtained by J. Ginibre and G. Velo, \cite{gv:scatter}.