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

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.


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


Zbl 1304.35132
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