No abstract

In this paper, we study the long time behavior of the solution of nonlinear Schrödinger equation with a singular potential. We prove scattering below the ground state for the radial NLS with inverse-square potential in dimension two iut + ∆u − au |x| 2 = −|u| p u when 2 < p < ∞ and a > 0. This work extends the result in [13,14,16] to dimension 2D. The key point is a modified version of Arora-Dodson-Murphy's approach [2].

No abstract

We study the restriction estimates in a class of conical singular space X = C(Y ) = (0, ∞)r × Y with the metric g = dr 2 + r 2 h, where the cross section Y is a compact (n − 1)-dimensional closed Riemannian manifold (Y, h). Let ∆g be the Friedrich extension positive Laplacian on X, and consider the operatorIn the present paper, we prove a type of modified restriction estimates for the solutions of wave equation associated with L V . The smallest positive eigenvalue of the operator ∆ h + V 0 + (n − 2) 2 /4 plays an important role in the result.As an application, for independent of interests, we prove local energy estimates and Keel-Smith-Sogge estimates for the wave equation in this setting.

No abstract