For the 2D inhomogeneous NLS with 0<b<1 and α>2-b, radial H^1 solutions scatter in both focusing (below ground state) and defocusing cases.
Scattering below the ground state for the 2$d$ radial nonlinear Schr\"odinger equation
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abstract
We revisit the problem of scattering below the ground state threshold for the mass-supercritical focusing nonlinear Schr\"odinger equation in two space dimensions. We present a simple new proof that treats the case of radial initial data. The key ingredient is a localized virial/Morawetz estimate; the radial assumption aids in controlling the error terms resulting from the spatial localization.
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2019 1verdicts
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Energy scattering for a class of inhomogeneous nonlinear Schr\"odinger equation in two dimensions
For the 2D inhomogeneous NLS with 0<b<1 and α>2-b, radial H^1 solutions scatter in both focusing (below ground state) and defocusing cases.