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We consider the energy-critical semilinear focusing wave equation indimension $N=3,4,5$. An explicit solution $W$ of this equation is known. By thework 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 $\|\nabla u_0\|_{L^2}<\|\nabla W\|_{L^2}$ isdefined globally and has finite $L^{\frac{2(N+1)}{N-2}}_{t,x}$-norm, whichimplies 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 acertain level of energy below $E(W,0)$ blows-up logarithmically as this levelapproaches the critical value $E(W,0)$. We also give a similar result in thecase of the radial energy-critical focusing semilinear Schr\"odinger equation.The proofs rely on the compactness argument of C. Kenig and F. Merle, on aclassification result, due to the authors, at the energy level $E(W,0)$, and onthe analysis of the linearized equation around $W$.

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