edoc: No conditions. Results ordered -Date Deposited. 2024-07-22T13:16:50ZEPrintshttps://edoc.unibas.ch/images/uni-logo.jpghttps://edoc.unibas.ch/2019-03-28T09:51:51Z2019-05-07T15:17:56Zhttps://edoc.unibas.ch/id/eprint/69988This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/699882019-03-28T09:51:51ZBlowup for Biharmonic NLSWeconsidertheCauchyproblemforthebiharmonic(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. Thomas BoulengerEnno Lenzmann2019-03-28T09:51:51Z2019-05-06T23:11:31Zhttps://edoc.unibas.ch/id/eprint/69989This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/699892019-03-28T09:51:51ZBlowup for fractional NLSWe consider fractional NLS with focusing power-type nonlinearity \[ i\partial_t u = (-\Delta)^s u - |u|^{2\sigma}u, (t,x) \in \mathbb{R}\times\mathbb{R}^N, \] where $1/2 < s < 1$ and $0 < \sigma < \infty$ for $s\ge N/2$ and $0 < \sigma \le 2s/(N-2s)$ for $s < N/2$. We prove a general criterion for blowup of radial solutions in $\mathbb{R}^N$ with $N\ge 2$ for $L^2$-supercritical and $L^2$-critical powers $\sigma\ge 2s/N$. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain $\Omega\subset\mathbb{R}^N$ in any dimension $N\ge 1$ and subject to exterior Dirichlet conditions. In this setting, we prove a general blowup result without imposing any symmetry assumption on u(t,x). For the blowup proof in $\mathbb{R}^N$, we derive a localized virial estimate for fractional NLS in $\mathbb{R}^N$ , which uses Balakrishnan's formula for the fractional Laplacian $(-\Delta)^s$ from semigroup theory. In the setting of bounded domains, we use a Pohozaev-type estimate for the fractional Laplacian to prove blowup. Thomas BoulengerDominik HimmelsbachEnno Lenzmann2018-08-14T13:39:59Z2018-08-14T13:39:59Zhttps://edoc.unibas.ch/id/eprint/40307This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/403072018-08-14T13:39:59ZBlowup for Biharmonic NLSWe 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. Thomas BoulengerEnno Lenzmann