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Register a new userSharp scattering threshold for the cubic-quintic NLS in the focusing-focusing regime
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Yongming Luo
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We consider the large data scattering problem for the 2D and 3D cubic-quintic NLS in the focusing-focusing regime. Our attention is firstly restricted to the 2D space, where the cubic nonlinearity is $L^2$-critical. We establish a new type of scattering criterion that is uniquely determined by the mass of the initial data, which differs from the classical setting based on the Lyapunov functional. At the end, we also formulate a solely mass-determining scattering threshold for the 3D cubic-quintic NLS in the focusing-focusing regime.
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We consider the nonlinear Schrodinger equation in three space dimensions with combined focusing cubic and defocusing quintic nonlinearity. This problem was considered previously by Killip, Oh, Pocovnicu, and Visan, who proved scattering for the whole region of the mass/energy plane where the virial quantity is guaranteed to be positive. In this paper we prove scattering in a larger region where the virial quantity is no longer guaranteed to be sign definite.
We adapt the arguments in the recent work of Duyckaerts, Landoulsi, and Roudenko to establish a scattering result at the sharp threshold for the $3d$ focusing cubic NLS with a repulsive potential. We treat both the case of short-range potentials as previously considered in the work of Hong, as well as the inverse-square potential, previously considered in the work of the authors.
In this article, we consider the focusing cubic nonlinear Schrodinger 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 Murphys 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.
This article resolves some errors in the paper Scattering threshold for the focusing nonlinear Klein-Gordon equation, Analysis & PDE 4 (2011) no. 3, 405-460. The errors are in the energy-critical cases in two and higher dimensions.
We study the construction of the Gibbs measures for the {it focusing} mass-critical fractional nonlinear Schrodinger equation on the multi-dimensional torus. We identify the sharp mass threshold for normalizability and non-normalizability of the focusing Gibbs measures, which generalizes the influential works of Lebowitz-Rose-Speer (1988), Bourgain (1994), and Oh-Sosoe-Tolomeo (2021) on the one-dimensional nonlinear Schrodinger equations. To this purpose, we establish an almost sharp fractional Gagliardo-Nirenberg-Sobolev inequality on the torus, which is of independent interest.
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