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

TL;DR

This paper investigates how the scattering norm for solutions to the energy-critical focusing wave equation behaves near the energy threshold, revealing a logarithmic blow-up as the energy approaches the critical level.

Contribution

It establishes the logarithmic blow-up rate of the scattering norm near the critical energy level for the focusing wave and Schrödinger equations, extending previous compactness and classification results.

Findings

Scattering norm blows up logarithmically near the energy threshold.

The blow-up rate is similar for wave and Schrödinger equations.

Results rely on linearized analysis around the explicit solution W.

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