BLOWUP FOR BIHARMONIC NLS

We consider the Cauchy problem for the biharmonic (i.e., fourth-order) NLS with focusing nonlinearity given by i partial derivative(t)u = Delta(2)u - mu Delta u - vertical bar u vertical bar(2 sigma)u for (t,x) is an element of [0, T) x R-d, where 0 < sigma < infinity for d 4 and 0 < sigma <= 4/(d - 4) for >= 5; and mu is an element of 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 (R-d) in any dimension >= 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 a = 4/d, we prove a general blowup result in finite or infinite time for radial data in H-2 (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.