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We consider the energy-critical semilinear focusing wave equation in dimension $N=3,4,5$. An explicit solution $W$ of this equation is known. By the work of C. Kenig and F. Merle, any solution of initial condition $(u_0,u_1)$ such that $E(u_0,u_1)<E(W,0)$ and $| abla u_0|_{L^2}<| abla W|_{L^2}$ is defined globally and has finite $L^{frac{2(N+1)}{N-2}}_{t,x}$-norm, which implies that it scatters. In this note, we show that the supremum of the $L^{frac{2(N+1)}{N-2}}_{t,x}$-norm taken on all scattering solutions at a certain level of energy below $E(W,0)$ blows-up logarithmically as this level approaches the critical value $E(W,0)$. We also give a similar result in the case of the radial energy-critical focusing semilinear Schrodinger equation. The proofs rely on the compactness argument of C. Kenig and F. Merle, on a classification result, due to the authors, at the energy level $E(W,0)$, and on the analysis of the linearized equation around $W$.
Consider a finite energy radial solution to the focusing energy critical semilinear wave equation in 1+4 dimensions. Assume that this solution exhibits type-II behavior, by which we mean that the critical Sobolev norm of the evolution stays bounded o
We extend the scattering result for the radial defocusing-focusing mass-energy double critical nonlinear Schrodinger equation in $dleq 4$ given by Cheng et al. to the case $dgeq 5$. The main ingredient is a suitable long time perturbation theory whic
We consider the energy-critical non-linear focusing wave equation in dimension N=3,4,5. An explicit stationnary solution, $W$, of this equation is known. The energy E(W,0) has been shown by C. Kenig and F. Merle to be a threshold for the dynamical be
Consider the energy-critical focusing wave equation in space dimension $Ngeq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. It is conjectured (soliton resolution) that any radial, bounded
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.