Scattering Norm Estimate Near the Threshold for Energy-critical Focusing Semilinear Wave Equation

We consider the energy-critical sermlinear 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 parallel to del u(0)parallel to(L2) < parallel to del W parallel to(L2) is defined globally and has finite L(t,x)((2(N+1))/(N-2))-norm, which implies that it scatters. In this note, we show that the supremum of the L(t,x)((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 Schrodinger equation. The proofs rely on the compactness argument of C. Kenig and E 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.