Blow-up for the 1D nonlinear Schr\"odinger equation with point nonlinearity I: Basic theory

Abstract

We consider the 1D nonlinear Schr\"odinger equation (NLS) with focusing point nonlinearity, $$ (\delta\text{NLS}) \qquad i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0, $$ where $\delta=\delta(x)$ is the delta function supported at the origin. We show that $\delta$NLS shares many properties in common with those previously established for the focusing autonomous translationally-invariant NLS $$ (\text{NLS}) \qquad i\partial_t \psi + \Delta \psi + |\psi|^{p-1}\psi=0 \,. $$ The critical Sobolev space $\dot H^{\sigma_c}$ for $\delta$NLS is $\sigma_c=\frac12-\frac{1}{p-1}$, whereas for NLS it is $\sigma_c=\frac{d}{2}-\frac{2}{p-1}$. In particular, the $L^2$ critical case for $\delta$NLS is $p=3$. We prove several results pertaining to blow-up for $\delta$NLS that correspond to key classical results for NLS. Specifically, we (1) obtain a sharp Gagliardo-Nirenberg inequality analogous to Weinstein (1983), (2) apply the sharp Gagliardo-Nirenberg inequality and a local virial identity to obtain a sharp global existence/blow-up threshold analogous to Weinstein (1983), Glassey (1977) in the case $\sigma_c=0$ and Duyckaerts, Holmer, & Roudenko (2008), Guevara (2014), and Fang, Xie, & Cazenave (2011) for $0<\sigma_c<1$, (3) prove a sharp mass concentration result in the $L^2$ critical case analogous to Tsutsumi (1990), Merle & Tsutsumi (1990) and (4) show that minimal mass blow-up solutions in the $L^2$ critical case are pseudoconformal transformations of the ground state, analogous to Merle (1993).