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, (0.1) i∂tψ + ∂ 2 xψ + δ|ψ|p−1ψ = 0, where δ = δ(x) is the delta function supported at the origin. In this work, we show that (0.1) shares many properties in common with those previously established for the focusing autonomous translationally-invariant NLS (0.2) i∂tψ + ∆ψ + |ψ|p−1ψ = 0 . The critical Sobolev space Ḣc for (0.1) is σc = 1 2 − 1 p−1 , whereas for (0.2) it is σc = d 2 − 2 p−1 . In particular, the L 2 critical case for (0.1) is p = 3. We prove several results pertaining to blow-up for (0.1) that correspond to key classical results for (0.2). Specifically, we (1) obtain a sharp Gagliardo-Nirenberg inequality analogous to Weinstein [Wei83], (2) apply the sharp Gagliardo-Nirenberg inequality and a local virial identity to obtain a sharp global existence/blow-up threshold analogous to Weinstein [Wei83], Glassey [Gla77] in the case σc = 0 and Duyckaerts, Holmer, & Roudenko [DHR08], Guevara [Gue14], and Fang, Xie, & Cazenave, [FXC11] for 0 < σc < 1, (3) prove a sharp mass concentration result in the L 2 critical case analogous to Tsutsumi [Tsu90], Merle & Tsutsumi [MT90] and (4) show that minimal mass blow-up solutions in the L critical case are pseudoconformal transformations of the ground state, analogous to Merle [Mer93].