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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].

We revisit the scattering results of the radial solutions below the ground state to the focusing inhomogeneous nonlinear Schrödinger equationwhen 0 < b < 1 and d = 2 in the 2D case. We use a modified version of Arora-Dodson-Murphy's approach [1] to give a new proof that extends the scattering results of [2] and avoids concentration compactness.

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In this paper, we give a new proof of the scattering and blow-up theory of the two coupled nonlinear Schrödinger system via establishing the corresponding interaction Morawetz estimate and scattering criterion. The method of this paper simplifies the proof in Xu, and the result of the paper improves the result in Xu. KEYWORDS blowup, coupled NLS system, Morawetz estimate, scattering MSC CLASSIFICATION 35L70; 35Q55