Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory

We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity, $(δ\text{NLS}) \qquad i\partial_tψ+ \partial_x^2ψ+ δ|ψ|^{p-1}ψ= 0,$ where $δ=δ(x)$ is the delta function supported at the origin. We show that $δ$NLS shares many properties in common with those previously established for the focusing autonomous translationally-invariant NLS $(\text{NLS}) \qquad i\partial_t ψ+ Δψ+ |ψ|^{p-1}ψ=0 \,.$ The critical Sobolev space $\dot H^{σ_c}$ for $δ$NLS is $σ_c=\frac12-\frac{1}{p-1}$, whereas for NLS it is $σ_c=\frac{d}{2}-\frac{2}{p-1}$. In particular, the $L^2$ critical case for $δ$NLS is $p=3$. We prove several results pertaining to blow-up for $δ$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 $σ_c=0$ and Duyckaerts, Holmer, & Roudenko (2008), Guevara (2014), and Fang, Xie, & Cazenave (2011) for $0<σ_c<1$, (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).

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