A minimal mass blow-up solution on a nonlinear quantum star graph

Abstract

We construct a finite-time blow-up solution to the mass-critical focusing nonlinear Schrödinger equation on a metric star graph with an arbitrary number of edges. We show that all solutions are global if their mass is smaller than an explicit constant, called "minimal mass". We then construct a solution with minimal mass and arbitrary energy, which blows up in finite time at the vertex of the star graph. The blow-up profile and blow-up speed are explicitly characterized. The main novelty of the paper is the construction of the blow-up profile in time-dependent domains of singularly perturbed Laplacians.

54 pages. Substantial revision to the proof of the main theorem. Specifically, the solution we construct belongs to the domain of the linearized operator and to suitable weighted Sobolev spaces. This has been made more explicit throughout the proofs