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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Abstract
We consider the Cauchy problem for the biharmonic (i.e., fourthorder) 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 Rd, 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 lowerorder dispersion. In the masssupercritical case sigma > 4/d, we prove a general result on finitetime blowup for radial data in H2 (Rd) 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 masscritical case a = 4/d, we prove a general blowup result in finite or infinite time for radial data in H2 (Rd). 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 
Publisher:  arXiv 
ISSN:  00129593 
eISSN:  18732151 
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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