Abstract. Consider the energy-critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions.Let W be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially the sum of a rescaled W concentrating at the origin and a small remainder which is continuous with respect to the time variable in the energy space. This is coherent with the solutions constructed by Krieger, Schlag and Tataru. One ingredient of our proof is that the unique radial solution which is compact up to scaling is equal to W up to symmetries.

Abstract. Scattering of radial H 1 solutions to the 3D focusing cubic nonlinear Schrö-dinger equation below a mass-energy thresholdwhere Q is the ground state, was established in Holmer-Roudenko [7]. In this note, we extend the result in [7] to non-radial H 1 data. For this, we prove a non-radial profile decomposition involving a spatial translation parameter. Then, in the spirit of Kenig-Merle [10], we control via momentum conservation the rate of divergence of the spatial translation parameter and by a convexity argument based on a local virial identity deduce scattering. An application to the defocusing case is also mentioned.

In this paper, we describe the asymptotic behaviour of globally defined solutions and of bounded solutions blowing up in finite time of the radial energy-critical focusing non-linear wave equation in three space dimension.

Let S(t) be a bounded strongly continuous semi-group on a Banach space B and −A be its generator. We say that S(t) is semi-uniformly stable when S(t)(A + 1) −1 tends to 0 in operator norm. This notion of asymptotic stability is stronger than pointwise stability, but strictly weaker than uniform stability, and generalizes the known logarithmic, polynomial and exponential stabilities.In this note we show that if S is semi-uniformly stable then the spectrum of A does not intersect the imaginary axis. The converse is already known, but we give an estimate on the rate of decay of S(t)(A + 1) −1 , linking the decay to the behaviour of the resolvent of A on the imaginary axis. This generalizes results of Lebeau and Burq (in the case of logarithmic stability) and Liu-Rao and Bátkai-Engel-Prüss-Schnaubelt (in the case of polynomial stability). BackgroundConsider a strongly continuous semi-group S(t) on a Banach space B, with generator −A. Assume that S(t) = e −tA is bounded, i.e. sup t≥0 e −tA = C < ∞.(1.1) (Throughout the article, a semi-group will be strongly continuous on [0, ∞), i.e., a C 0 -semigroup. Moreover, · will denote both the norm on B and the operator norm from B to B.) The operator A is closed and densely defined, and we denote by D(A) its domain, σ(A) its spectrum and ρ(A) its resolvent set. It is a well-known property that if (1.1) holds, then the left open half-plane {Re z < 0} is included in ρ(A) (see [25,11]). In 1988, Lyubich and Vũ [20] and Arendt and Batty [1] have shown that if σ(A) ∩ iR is countable and σ(A * ) ∩ iR contains no eigenvalue (here A * is the adjoint of A), then the semi-group is (pointwise) strongly stable, that ist→+∞ e −tA u 0 = 0. For surveys of this and other results concerning strong stability, see [4], [8] 1 .This work was partially supported by the French ANR ControlFlux. The second author would like to thank Nicolas Burq for fruitful discussions on the subject, and Luc Miller for pointing out the stability theorem of Lyubich, Vũ, Arendt and Batty and the article [3]. 1 We will only address strong stability concepts, and refer to [10] for a recent survey on different types of weak stability.

Abstract. We consider the energy-critical non-linear focusing Schrödinger equation in dimension N = 3, 4, 5. An explicit stationnary solution, W , of this equation is known. In [KM06], the energy E(W ) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article, we study the dynamics at the critical level E(u) = E(W ) and classify the corresponding solutions. This gives in particular a dynamical characterization of W .

We consider the energy-critical non-linear focusing wave equation in dimension N = 3, 4, 5. An explicit stationnary solution, W , of this equation is known. In [KM06b], the energy E(W, 0) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article we study the dynamics at the critical level E(u0, u1) = E(W, 0) and classify the corresponding solutions. We show in particular the existence of two special solutions, connecting different behaviors for negative and positive times. Our results are analoguous to [DM07], which treats the energy-critical non-linear focusing radial Schrödinger equation, but without any radial assumption on the data. We also refine the understanding of the dynamical behavior of the special solutions.

We study the focusing 3d cubic NLS equation with H 1 data at the mass-energy threshold, namely, when. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) was classified when. In this paper, we first exhibit 3 special solutions: e it Q and Q ± , where Q is the ground state, Q ± exponentially approach the ground state solution in the positive time direction, Q + has finite time blow up and Q − scatters in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up toḢ 1/2 symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold. These results are obtained by studying the spectral properties of the linearized Schrödinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.

Consider a finite energy radial solution to the focusing energy critical semilinear wave equation in 1+4 dimensions. Assume that this solution exhibits type-II behavior, by which we mean that the critical Sobolev norm of the evolution stays bounded on the maximal interval of existence. We prove that along a sequence of times tending to the maximal forward time of existence, the solution decomposes into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume that the critical norm of the evolution localized to the light cone (the forward light cone in the case of global solutions and the backwards cone in the case of finite time blow-up) is less than 2 times the critical norm of the ground state solution W , then the decomposition holds without a restriction to a subsequence.