Blowup for Biharmonic NLS

Boulenger, Thomas and Lenzmann, Enno. (2015) Blowup for Biharmonic NLS. Annales scientifiques de l'ENS - Parutions - série 4, 50 (3).

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Official URL: http://edoc.unibas.ch/40307/

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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.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Lenzmann)
UniBasel Contributors:Lenzmann, Enno and Boulenger, Thomas
Item Type:Working Paper
Note:Publication type according to Uni Basel Research Database: Discussion paper / Internet publication
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Last Modified:14 Aug 2018 13:39
Deposited On:14 Aug 2018 13:39

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