Blowup for Biharmonic NLS

Boulenger, Thomas and Lenzmann, Enno. (2015) Blowup for Biharmonic NLS. Preprints Fachbereich Mathematik, 2015 (17).

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Official URL: https://edoc.unibas.ch/69988/

$i\partial_t u = \Delta^2 u - \mu \Delta u - |u|^{\sigma 2} u for (t,x) \in [0,T)\times \mathbb{R}^d,$
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.