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.

中文翻译：

---

凸障碍物外部非线性薛定谔方程的爆破解

在本文中，我们在具有狄利克雷边界条件的光滑、紧凑、凸形障碍物 $\mathbb{R}^d$ 的外部考虑具有质量超临界聚焦非线性的薛定谔方程。我们证明负能量的解决方案在有限的时间内爆炸。进一步假设非线性是能量次临界的，我们还证明（在额外的对称条件下）爆炸具有与 Holmer 和 Roudenko 在 $\mathbb{R}^d$ 上的工作相同的最佳基态标准。Glassey 的经典证明基于方差的凹性，在障碍物的外部失败，因为在方差的二阶导数中出现了带有不利符号的边界项。