Blowup of cylindrically symmetric solutions for biharmonic NLS

Abstract

In this paper, we consider blowup of solutions to the Cauchy problem for the following biharmonic nonlinear Schrödinger equation (NLS), $$\text{i}{\partial }_{t}u={\Delta }^{2}u-\mu \Delta u-|u{|}^{2\sigma }u\text{in}\mathbb{R}\times {\mathbb{R}}^d,$$ where $d\ge 1$, $\mu \in \mathbb{R}$ and $0<\sigma <\infty$ if $1\le d\le 4$ and $0<\sigma <4/(d-4)$ if $d\ge 5$. 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.

10 pages

References in corpus (3)

• Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case
• Ring-type singular solutions of the biharmonic nonlinear Schrodinger equation
• On finite time blow-up for a 3D Davey-Stewartson system