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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
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