Blowup of cylindrically symmetric solutions for biharmonic NLS

TL;DR

This paper investigates the blowup behavior of cylindrically symmetric solutions to a biharmonic nonlinear Schrödinger equation, extending previous results from radial symmetry to a broader class of symmetric solutions.

Contribution

It establishes the existence of blowup solutions for cylindrically symmetric data in mass critical and supercritical regimes, expanding the understanding beyond radial symmetry cases.

Findings

Blowup solutions exist for cylindrically symmetric initial data.

Results extend previous blowup results from radial to cylindrical symmetry.

Applicable in mass critical and supercritical cases for certain dimensions.

Abstract

In this paper, we consider blowup of solutions to the Cauchy problem for the following biharmonic nonlinear Schr\"odinger equation (NLS), $$ \textnormal{i} \, \partial_t u=\Delta^2 u-\mu \Delta u-|u|^{2 \sigma} u \quad \text{in} \,\, \R \times \R^d, $$ where $d \geq 1$, $\mu \in \R$ and $0<\sigma<\infty$ if $1 \leq d \leq 4$ and $0<\sigma<4/(d-4)$ if $d \geq 5$. In the mass critical and supercritical cases, we establish the existence of blowup solutions to the problem for cylindrically symmetric data. The result extends the known ones with respect to blowup of solutions to the problem for radially symmetric data.