Abstract
In this article, we consider the focusing cubic nonlinear Schrödinger equation(NLS) in the exterior domain outside of a convex obstacle in ℝ3 with Dirichlet boundary conditions. We revisit the scattering result below ground state in Killip—Visan—Zhang [The focusing cubic NLS on exterior domains in three dimensions. Appl. Math. Res. Express. AMRX, 1, 146–180 (2016)] by utilizing the method of Dodson and Murphy [A new proof of scattering below the ground state for the 3d radial focusing cubic NLS. Proc. Amer. Math. Soc., 145, 4859–4867 (2017)] and the dispersive estimate in Ivanovici and Lebeau [Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples. Comp. Rend. Math., 355, 774–779 (2017)], 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 that the solution decays at a large time interval, which meets the scattering criterion.
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Acknowledgements
We thank the referees for their time and comments. And we would also like to thank Jason Murphy for his helpful discussions.
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Supported by Beijing Natural Science Foundation 1222019, PFCAEP project (Grant No. YZJJLX201901), NSFC (Grant No. 11901041, 12101040) and NSAF (Grant No. U1530401)
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Xu, C.B., Zhao, T.F. & Zheng, J.Q. Scattering for 3D Cubic Focusing NLS on the Domain Outside a Convex Obstacle Revisited. Acta. Math. Sin.-English Ser. 38, 1054–1068 (2022). https://doi.org/10.1007/s10114-022-1058-x
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DOI: https://doi.org/10.1007/s10114-022-1058-x