Blow-up for the pointwise NLS in dimension two: Absence of critical power

Abstract

We consider the Schrödinger equation in dimension two with a fixed, pointwise, focusing nonlinearity and show the occurrence of a blow-up phenomenon with two peculiar features: first, the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves. Second, there is no critical power nonlinearity, i.e. for every power there exist blow-up solutions. This last property is uncommon among the conservative Schrödinger equations with local nonlinearity.

Scattering for the 𝐿² supercritical point NLS

“…Higher-dimensional models with a generalization of the delta potential have been introduced in [2] and in [6] for the three and two-dimensional setting, respectively. While, at a qualitative level, the model in dimension three behaves like that in dimension one, the two-dimensional setting displays some uncommon features still to be understood (for the analysis of the blow-up, see [1]).…”

Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one

“…On the contrary, when s P p 1 4 , 1 2 s, the issue presents the structure of higher codimensional models, such as nonlinear δ-perturbations of the Laplacian in R 2 and R 3 (precisely, s P p 1 4 , 1 2 q retraces delta in 3d, while s " 1 2 retraces delta in 2d). In these cases the strategy for the proof of global and local well-posedness (and blow-up) is considerably different (for instance, it is no more true that qptq " ψpt, 0q) and, mainly in the 2d case ( [2,14]), requires a more refined analysis of the kernel of the associated charge equation, which is not clear how to adapt to the fractional case at the moment. Hence, we think that in a first work on nonlinear perturbations of the fractional Laplacian, it is worth starting from one-codimensional problems, leaving higher-codimensional ones to forthcoming papers.…”

“…The major characteristic of this kind of equations is that, albeit nonlinear, they fall under the so-called solvable models: the investigation of the time evolution can be reduced to that of an ODE-type equation (see, e.g., [6]). For the ordinary Laplacian in R the case of a concentrated nonlinearity has been studied in [5] (see also [15]) and then extended to R 2 and R 3 in [2,13,14,16] and [3,4] (respectively), and to the 1-dimensional Dirac equation in [10].…”

On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction

“…As related topics, we mention the NLS with the concentrated nonlinearity. See [3,17,42] for N = 1, [1,2] for N = 2, and [5] for N = 3.…”