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Mathematics > Analysis of PDEs

arXiv:1812.09445 (math)
[Submitted on 22 Dec 2018]

Title:Scattering for 3d cubic focusing NLS on the domain outside a convex obstacle revisited

Authors:Chengbin Xu, Tengfei Zhao, Jiqiang Zheng
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Abstract:In this article, we consider the focusing cubic nonlinear Schrödinger equation(NLS) in the exterior domain outside of a convex obstacle in $\mathbb{R}^3$ with Dirichlet boundary conditions. We revisit the scattering result below ground state of Killip-Visan-Zhang by utilizing Dodson and Murphy's argument and the dispersive estimate established by Ivanovici and Lebeau, which avoids using the concentration compactness. We conquer the difficulty of the boundary in the focusing case by establishing a local smoothing effect of the boundary. Based on this effect and the interaction Morawetz estimates, we prove the solution decays at a large time interval, which meets the scattering criterions.
Comments: 15 pages, comments are welcome
Subjects: Analysis of PDEs (math.AP)
Cite as: arXiv:1812.09445 [math.AP]
  (or arXiv:1812.09445v1 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.1812.09445

Submission history

From: Jiqiang Zheng [view email]
[v1] Sat, 22 Dec 2018 03:51:10 UTC (22 KB)
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