  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 $\|\nabla u_0\|_{L^2}<\|\nabla W\|_{L^2}$ 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 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 $W$.
