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

Abstract

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 ∥∇u 0 ∥ L 2 < ∥∇W∥ L 2 is defined globally and has finite L (2(N+1))/(N-2) t,x -norm, which implies that it scatters. In this note, we show that the supremum of the L (2(N+I))/(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.