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We consider the Cauchy problem for the biharmonic (i. e. fourth-order) NLS with focusing nonlinearity given bywhere 0 ă σ ă 8 for d ď 4 and 0 ă σ ď 4{pd´4q for d ě 5; and µ P R is some parameter to include a possible lower-order dispersion. In the mass-supercritical case σ ą 4{d, we prove a general result on finite-time blowup for radial data in H 2 pR d q in any dimension d ě 2. Moreover, we derive a universal upper bound for the blowup rate for suitable 4{d ă σ ă 4{pd´4q. In the mass-critical case σ " 4{d, we prove a general blowup result in finite or infinite time for radial data in H 2 pR d q. 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.

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