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 $\mathrm{\parallel }\mathrm{\nabla }{u}_0{\mathrm{\parallel }}_{L^2}<\mathrm{\parallel }\mathrm{\nabla }W{\mathrm{\parallel }}_{L^2}$ is defined globally and has finite $L_{t,x}^{\frac{2(N+1)}{N-2}}$-norm, which implies that it scatters. In this note, we show that the supremum of the $L_{t,x}^{\frac{2(N+1)}{N-2}}$-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 Schr"odinger 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 $W$.