Asymmetric minimal mass blow-up for the quintic NLS on a star graph

Abstract

This work is concerned with the focusing quintic nonlinear Schrödinger equation on a star graph with two edges, with a delta interaction of arbitrary strength at the vertex. We construct a non-radial, minimal mass, finite time blow-up solution, whose singularity forms at an arbitrarily prescribed interior point of one edge. Our approach is based on a suitable cut-off of the standard pseudoconformal blow-up solution of the focusing quintic NLS on the line, inspired by Ogawa-Tsutsumi and Burq-Gérard-Tzvetkov. A noteworthy feature of our construction is the persistence of the domain of the Hamiltonian under the flow, obtained by the persistence of regularity method of Kato. We also identify a two-parameter family of distinct blow-up profiles, which should play a central role in the classification of non-radial minimal mass blow-up solutions on the star graph. Our construction is generic and can be extended to a star graph with an arbitrary number of edges. It can certainly also be adapted to other types of metric graphs.