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 consider the global dynamics below the ground state energy for the Zakharov system in the 3D radial case. We obtain dichotomy between the scattering and the growup.
We prove small energy scattering for the 3D Zakharov system with radial symmetry. The main ingredients are normal form reduction and the radial-improved Strichartz estimates.
Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^infty$. It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modifie
