The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation

Abstract

We consider the critical nonlinear Schrödinger equation iu t = −∆u−|u| 4 N u with initial condition u(0, x) = u 0 in dimension N = 1. For u 0 ∈ H 1 , local existence in the time of solutions on an interval [0, T ) is known, and there exist finite time blow-up solutions, that is, u 0 such that lim t↑T <+∞ |u x (t)| L 2 = +∞. This is the smallest power in the nonlinearity for which blow-up occurs, and is critical in this sense. The question we address is to understand the blow-up dynamic. Even though there exists an explicit example of blow-up solution and a class of initial data known to lead to blow-up, no general understanding of the blow-up dynamic is known. At first, we propose in this paper a general setting to study and understand small, in a certain sense, blow-up solutions. Blow-up in finite time follows for the whole class of initial data in H 1 with strictly negative energy, and one is able to prove a control from above of the blow-up rate below the one of the known explicit explosive solution which has strictly positive energy. Under some positivity condition on an explicit quadratic form, the proof of these results adapts in dimension N > 1.

Why are solitons stable?

“…These results do not quite follow exactly the same pattern as the results for critical gKdV equations mentioned in the previous section, but certainly share many of the same ingredients. For instance, see [60], [61], [62] for some results relating to the mass-critical NLS. For the energycritical nonlinear wave equation or wave maps equation, there are some slightly different approaches to create blowup [34], [37], [71]; see [72] for a recent survey.…”

A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

“…In contrast, for the critical exponent p * = 2 d , global existence of a solution u of (1.1) is guaranteed only under a smallness condition, elaborated in equation (1.3). If this smallness condition does not hold, blowup solutions of problem (1.1) exist [40,41,42].…”

“…e.g., [4,10,17,40,41,42,44,50] and the references therein. Moreover 2 THEODOROS KATSAOUNIS AND IRENE KYZA this activity includes qualitative and asymptotic questions, cf.…”

“…Similar estimates are used in the a priori analysis for establishing existence of solutions for (1.1), cf. [26,41].…”

“…If λ > 0, problem (1.1) is called the focusing NLS equation. In this case, if (1.3) is not satisfied, i.e., if Γ(u 0 ) ≥ 1, the solution u of (1.1) may blow up in the H 1 −norm in some finite time T * < ∞ (cf., e.g., [40,41,42] and the references therein). The focusing cases we consider in this paper are those with Γ(u 0 ) < 1, thus a global solution of (1.1) exists.…”