论文标题
在凸障碍物外部的非线性schrödinger方程的爆破解决方案上
On Blow-up solutions to the nonlinear Schrödinger equation in the exterior of a convex obstacle
论文作者
论文摘要
在本文中，我们考虑了具有质量质量焦点的非线性的schrödinger方程，在平滑，紧凑，凸障碍的外部$ \ r^{d} $的外部，并具有dirichlet边界条件。我们证明，在有限的时间内，具有负能量的解决方案爆炸。此外，假设非线性是能量 - 关键的，我们还证明了（在其他对称条件下）与$ \ r^{d} $上的霍尔默（Holmer）和鲁登科（Roudenko）的工作相比，使用相同的最佳地面标准爆炸。基于方差的凹度，格拉斯西的经典证据在障碍物的外部失败，因为边界术语的出现在方差的第二个衍生物中具有不利的符号。我们证明的主要思想是引入一个新的修改差异，该方差是从下面限制的，并严格凹入了我们考虑的解决方案。
In this paper, we consider the Schrödinger 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.
