On blow-up solutions to the nonlinear Schrödinger equation in the exterior of a convex obstacle

In this paper, we consider the Schrödinger equation with a masssupercritical focusing nonlinearity, in the exterior of a smooth, compact, convex obstacle of $\mathbb{R}^d$ 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 $\mathbb{R}^d$. 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.

Keywords

focusing NLS equation, exterior domain, boundary value problem, blow-up

2010 Mathematics Subject Classification

Primary 35Q55. Secondary 35B40, 35B44, 35G30, 35K20, 58J32.

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Received 20 April 2021

Published 2 December 2021