Blowup for Biharmonic NLS

WeconsidertheCauchyproblemforthebiharmonic(i.e.fourth-order) NLS with focusing nonlinearity given by
\[
i\partial_t u = \Delta^2 u - \mu \Delta u - |u|^{\sigma 2} u for (t,x) \in [0,T)\times \mathbb{R}^d,
\]
where $0 < \sigma < \infty$ for $d\le 4$ and $0 < \sigma \le 4/(d-4)$ for $d \ge 5$; and $\mu\in\mathbb{R}$ is some parameter to include a possible lower-order dispersion. In the mass-supercritical case $\sigma > 4/d$ , we prove a general result on finite-time blowup for radial data in $H^2(\mathbb{R}^d)$ in any dimension $d \ge 4$. Moreover, we derive a universal upper bound for the blowup rate for suitable $4/d < \sigma < 4/(d-4)$. In the mass-critical case $\sigma = 4/d$, we prove a general blowup result in finite or infinite time for radial data in $H^2(\mathbb{R}^d)$. As a key ingredient, we utilize the time evolution of a nonnegative quantity, which we call the (localized) Riesz bivariance for biharmonic NLS. This construction provides us with a suitable substitute for the variance used for classical NLS problems.
In addition, we prove a radial symmetry result for ground states for the bihar- monic NLS, which may be of some value for the related elliptic problem.