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

TL;DR

This paper constructs a finite-time blow-up solution for the mass-critical focusing nonlinear Schrödinger equation on a star graph, explicitly characterizing the blow-up profile and speed, and identifying a minimal mass threshold for global solutions.

Contribution

It introduces a novel construction of blow-up solutions on quantum star graphs with explicit profiles and speeds, advancing understanding of singularity formation in such domains.

Findings

Solutions are global if mass is below a critical threshold.

A minimal mass blow-up solution is explicitly constructed.

The blow-up profile and speed are precisely characterized.

Abstract

We construct a finite-time blow-up solution to the mass-critical focusing nonlinear Schr\"odinger 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.