On Blow-up solutions to the nonlinear Schr\"odinger equation in the exterior of a convex obstacle

TL;DR

This paper proves finite-time blow-up of solutions to a nonlinear Schrödinger equation outside a convex obstacle, introducing a new variance method to handle boundary complications.

Contribution

It extends blow-up results for the nonlinear Schrödinger equation to exterior convex domains using a novel variance approach.

Findings

Solutions with negative energy blow up in finite time.

Blow-up occurs under energy-subcritical conditions with symmetry assumptions.

A new bounded, concave variance method overcomes boundary term challenges.

Abstract

In this paper, we consider the Schr\"odinger equation with a mass-supercritical focusing nonlinearity, in the exterior of a smooth, compact, convex obstacle of $\R^{d}$ with Dirichlet boundary conditions. We prove that solutions with negative energy blow up in finite time. Assuming furthermore that the nonlinearity is energy-subcritical, we also prove (under additional symmetry conditions) blow-up with the same optimal ground-state criterion than in the work of Holmer and Roudenko on $\R^{d}$. The classical proof of Glassey, based on the concavity of the variance, fails in the exterior of an obstacle because of the appearance of boundary terms with an unfavorable sign in the second derivative of the variance. The main idea of our proof is to introduce a new modified variance which is bounded from below and strictly concave for the solutions that we consider.