Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation

These spaces are natural for KP-II equation because of the following considerations: The homogeneous space H ˙ − 1 2 ,0 ( R 2 ) is invariant under the scaling symmetry (2) of

VELO, The global Cauchy problem for the nonlinear Klein-Gordon

ferential operators are discussed in section 3. Bounds and asymptotics of scattering phases are obtained in section 4... 2. PERTURBATION OF AN ISOLATED EIGENVALUE

If, in addition, we assume that the critical norm of the evolution localized to the light cone (the forward light cone in the case of global solutions and the backwards cone in the

The equations covered by our results include the 2 dimensional corotational wave map system, with target S 2 , in the critical energy space, as well as the 4 dimensional,

A similar result holds for the critical radial Yang-Mills equation (exponential concentration rate); the same would be the case for (WM) with k = 2. In any dimension N > 6 we

The wave equation has no new blow-up rate at the critical case, unlike NLS where new blow-up rates appear at the critical case (with respect to the conformal invariance), see Merle

Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schr¨ odinger equation in the radial case.. Kenig and