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We consider the critical nonlinear Schrödinger equation iu t = −∆u−|u| 4 N u with initial condition u(0, x) = u 0 in dimension N = 1. For u 0 ∈ H 1 , local existence in the time of solutions on an interval [0, T ) is known, and there exist finite time blow-up solutions, that is, u 0 such that lim t↑T <+∞ |u x (t)| L 2 = +∞. This is the smallest power in the nonlinearity for which blow-up occurs, and is critical in this sense. The question we address is to understand the blow-up dynamic. Even though there exists an explicit example of blow-up solution and a class of initial data known to lead to blow-up, no general understanding of the blow-up dynamic is known. At first, we propose in this paper a general setting to study and understand small, in a certain sense, blow-up solutions. Blow-up in finite time follows for the whole class of initial data in H 1 with strictly negative energy, and one is able to prove a control from above of the blow-up rate below the one of the known explicit explosive solution which has strictly positive energy. Under some positivity condition on an explicit quadratic form, the proof of these results adapts in dimension N > 1.

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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.

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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.