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

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 $W$.