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

We consider the 1D nonlinear Schrodinger equation (NLS) with focusing point nonlinearity, i partial derivative(t)Psi + partial derivative(2)(x)Psi + delta vertical bar Psi vertical bar(p-1)Psi = 0, (0.1) where delta = delta(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 i partial derivative(t)Psi + Delta Psi + vertical bar Psi vertical bar(p-1)Psi = 0. (0.2) The critical Sobolev space H-sigma c for (0.1) is sigma(c) = 1/2 - 1/p-1, whereas for (0.2) it is sigma(c) = 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 analogousto Weinstein [44], (2) apply the sharp Gagliardo-Nirenberg inequality and a local virial identity to obtain a sharp global existence/blow-up threshold analogous to Weinstein [44], Glassey [17] in the case sigma(c) = 0 and Duyckaerts, Holmer, & Roudenko [12], Guevara [18], and Fang, Xie, & Cazenave, [13] for 0 < sigma(c) < 1, (3) prove a sharp mass concentration result in the L-2 critical case analogous to Tsutsumi [43], Merle Tsutsumi [36] 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 [28]. (C) 2019 Published by Elsevier Inc.