论文标题
焦点3D立方NLS方程中的阈值解决方案在严格凸出障碍物之外
Threshold solutions in the focusing 3D cubic NLS equation outside a strictly convex obstacle
论文作者
论文摘要
我们研究了重点$ 3D $ CUTIC非线性schrödinger方程的动力学在严格凸出障碍物的外部质量能阈值的外部，也就是说，当$e_Ω[u_0]m_Ω[u_0] [u_0] [u_0] \ nabla u_0 \ right \ | _ {l^{2}（ω）}} \ left \ | u_0 \ right \ | _ {l^{2}（ω）} <\ left \ | \ nabla q \ right \ | _ {l^2（\ r^3）} \ left \ | q \ right \ | _ {l^2（\ r^3）}，$ $ u_0 \ in H^1_0（ω）$是初始数据，$ q $是欧几里得空间的基础状态，$ e $是能量，$ m $是质量。在整个Euclidean Space中，Duyckaerts和Roudenko（遵循Duyckaerts和Merle在临界问题上的工作）证明，存在一种特定的全球解决方案，该解决方案散布了负时代，并在积极的时间内散布了孤儿。我们证明，对于外部域中的问题，这些杂斜轨道不存在，并且阈值的所有解决方案都是全球定义和散射的。主要困难是控制空间转换参数，因为伽利略转换不可用。
We study the dynamics of the focusing $3d$ cubic nonlinear Schrödinger equation in the exterior of a strictly convex obstacle at the mass-energy threshold, namely, when $ E_Ω[u_0] M_Ω[u_0] = E_{\R^3}[Q] M_{\R^3}[Q] $ and $ \left\| \nabla u_0 \right\|_{L^{2}(Ω)} \left\|u_0\right\|_{L^{2}(Ω)}< \left\| \nabla Q \right\|_{L^2(\R^3)} \left\| Q \right\|_{L^2(\R^3)} ,$ where $u_0 \in H^1_0(Ω)$ is the initial data, $Q$ is the ground state on the Euclidean space, $E$ is the energy and $M$ is the mass. In the whole Euclidean space Duyckaerts and Roudenko (following the work of Duyckaerts and Merle on the energy-critical problem) have proved the existence of a specific global solution that scatters for negative times and converges to the soliton in positive times. We prove that these heteroclinic orbits do not exist for the problem in the exterior domain and that all solutions at the threshold are globally defined and scatter. The main difficulty is the control of the space translation parameter, since the Galilean transformation is not available.
