  We consider the Cauchy problem for the biharmonic (i.\,e.~fourth-order) NLS
with focusing nonlinearity given by $i \partial_t u = \Delta^2 u - \mu \Delta u
-|u|^{2 \sigma} u$ for $(t,x) \in [0,T) \times \mathbb{R}^d$, where $0 < \sigma
<\infty$ for $d \leq 4$ and $0 < \sigma \leq 4/(d-4)$ for $d \geq 5$; 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 $d
\geq 2$. 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 biharmonic NLS, which may be of some
value for the related elliptic problem.
