On dispersive blow-ups for the nonlinear Schrödinger equation. (English) Zbl 1413.35092

Hong, Younghun; Taskovic, Maja

Differ. Integral Equ. 29, No. 9-10, 875-888 (2016).

This article is a refinement and extension of J. L. Bona et al. [J. Math. Pures Appl. 102, No. 4, 782–811 (2014; Zbl 1304.35132)] about the construction of dispersive blow-up solutions to the nonlinear Schrödinger equation.
The basic idea in this paper and the quoted article is that it is sufficient to construct such a solution for the linear Schrödinger equation and then show that the nonlinear term can be considered as a smoother perturbation by the mean of Duhamel’s formula and thanks to (time) dispersive estimates.
Here the main contribution is a uniform pointwise estimate for the time integrated nonlinear term (in Duhamel’s formula) in term of the \(L^{\infty}_tH^s_x\) norm of the solution for \(s>\frac{d}{2}-\frac{2}{p}\) where \(p\) is the power of the nonlinear term (for comparison \(s>\frac{d}{2}-\frac{1}{2(p+1)}\) in the quoted article).
In addition to single point blow-up solutions, the authors provide dispersive blow-ups on a straight line and on a sphere.

Reviewer: Vincent Lescarret (Gif-sur-Yvette)

MSC:

35B44	Blow-up in context of PDEs
35Q55	NLS equations (nonlinear Schrödinger equations)
35L67	Shocks and singularities for hyperbolic equations
35L70	Second-order nonlinear hyperbolic equations

Keywords:

nonlinear Schrödinger equation; dispersive blow-up; smoothing effect

Citations:

Zbl 1304.35132