Sharp Thresholds of Instability for the Biharmonic Nonlinear Schrödinger Equations

We consider the time-dependent biharmonic nonlinear Schrödinger equations with and without a mixed dispersion. We find the sharp threshold of blow-up and global existence for the biharmonic nonlinear Schrödinger equations without a mixed dispersion by establishing the modified cross-constraint variational method. Building on this work, we prove that the standing waves are strongly unstable without further assumptions on the frequency. The main new ingredients of which require us to provide a comparison of two variational problems, improved the method in Zhang [Comm PDEs, 2005, 30: 1429-1443]. In the latter part of the paper, we study the biharmonic nonlinear Schrödinger equations with a mixed dispersion, which destroys the scaling invariance. The single variational approach is utilized to obtain the sharp threshold of blow-up and global existence, and strong instability of standing waves.