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Register a new userCubic-quintic NLS: scattering beyond the virial threshold
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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.
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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.
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 paper, we study the scattering theory for the cubic inhomogeneous Schrodinger equations with inverse square potential $iu_t+Delta u-frac{a}{|x|^2}u=lambda |x|^{-b}|u|^2u$ with $a>-frac14$ and $0<b<1$ in dimension three. In the defocusing case (i.e. $lambda=1$), we establish the global well-posedness and scattering for any initial data in the energy space $H^1_a(mathbb R^3)$. While for the focusing case(i.e. $lambda=-1$), we obtain the scattering for the initial data below the threshold of the ground state, by making use of the virial/Morawetz argument as in Dodson-Murphy [Proc. Amer. Math. Soc.,145(2017), 4859-4867.] and Campos-Cardoso [arXiv: 2101.08770v1.] that avoids the use of interaction Morawetz estimate.
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.
The present paper is concerned with the large data scattering problem for the mass-energy double critical NLS begin{align} ipartial_t u+Delta upm |u|^{frac{4}{d}}upm |u|^{frac{4}{d-2}}u=0tag{DCNLS} end{align} in $H^1(mathbb{R}^d)$ with $dgeq 3$. In the defocusing-defocusing regime, Tao, Visan and Zhang show that the unique solution of DCNLS is global and scattering in time for arbitrary initial data in $H^1(mathbb{R}^d)$. This does not hold when at least one of the nonlinearities is focusing, due to the possible formation of blow-up and soliton solutions. However, precise thresholds for a solution of DCNLS being scattering were open in all the remaining regimes. Following the classical concentration compactness principle, we impose sharp scattering thresholds in terms of ground states for DCNLS in all the remaining regimes. The new challenge arises from the fact that the remainders of the standard $L^2$- or $dot{H}^1$-profile decomposition fail to have asymptotically vanishing diagonal $L^2$- and $dot{H}^1$-Strichartz norms simultaneously. To overcome this difficulty, we construct a double track profile decomposition which is capable to capture the low, medium and high frequency bubbles within a single profile decomposition and possesses remainders that are asymptotically small in both of the diagonal $L^2$- and $dot{H}^1$-Strichartz spaces.
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