Registered users receive a variety of benefits including the ability to customize email alerts, create favorite journals list, and save searches.
Please note that a Project Euclid web account does not automatically grant access to full-text content. An institutional or society member subscription is required to view non-Open Access content.
Contact email@example.com with any questions.
We consider the quintic two-dimensional focusing nonlinear Schrödinger equation which is -supercritical. Even though the existence of finite-time blow-up solutions in the energy space is known, very little is understood concerning the singularity formation. Numerics suggest the existence of a stable blow-up dynamic corresponding to a self-similar blowup at one point in space. We prove the existence of a different type of dynamic and exhibit an open set among the -radial distributions of initial data for which the corresponding solution blows up in finite time on a sphere. This is the first result of an explicit description of a blow-up dynamic in the -supercritical setting
We consider Kapranov's Chow quotient compactification of the moduli space of ordered -tuples of hyperplanes in in linear general position. For , this is canonically identified with the Grothendieck-Knudsen compactification of which has, among others, the following nice properties:
We prove the existence of small amplitude, ()-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency belonging to a Cantor-like set of asymptotically full measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser implicit function theorem. In spite of the complete resonance of the equation, we show that we can still reduce the problem to a finite-dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows us to deal also with nonlinearities that are not odd and with finite spatial regularity
PURCHASE SINGLE ARTICLE
This article is only available to subscribers. It is not available for individual sale.