Blow-up of the non-equivariant 2+1 dimensional wave map

Renormalization and Blow up for Charge One Equivariant Critical Wave Maps

We prove the existence of equivariant finite time blow-up solutions for the wave map problem from R2+1 → S2 of the form u(t, r) = Q(λ(t)r)+R(t, r) where u is the polar angle on the sphere, Q(r) = 2 arctan r is the ground state harmonic map, λ(t) = t−1−ν , and R(t, r) is a radiative error with local energy going to zero as t→ 0. The number ν > 1 2 can be described arbitrarily. This is accomplish...

On the existence of smooth self-similar blow-up profiles for the wave-map equation

Consider the equivariant wave map equation from Minkowski space to a rotationnally symmetric manifold N which has an equator (example: the sphere). In dimension 3, this article presents a necessary and sufficient condition on N for the existence of a smooth self-similar blow up profile. More generally, we study the relation between the minimizing properties of the equator map for the Dirichlet ...

Smooth type II blow up solutions to the four dimensional energy critical wave equation

We exhibit C∞ type II blow up solutions to the focusing energy critical wave equation in dimension N = 4. These solutions admit near blow up time a decomposiiton u(t, x) = 1 λ N−2 2 (t) (Q+ ε(t))( x λ(t) ) with ‖ε(t), ∂tε(t)‖Ḣ1×L2 ≪ 1 where Q is the extremizing profile of the Sobolev embedding Ḣ → L∗ , and a blow up speed λ(t) = (T − t)e− √ |log(T−t)|(1+o(1)) as t → T.

Capacity and blow - up for the 3 + 1 dimensional wave operator

Blow up for the Semilinear Wave Equation in Schwarzschild Metric

We study the semilinear wave equation in Schwarzschild metric (3 + 1 dimensional space time). First, we establish that the problem is locally well posed in H for any σ > 1; then we prove the blow up of the solution for every p > 1 and non negative initial data. The work is dedicated to prof. Yvonne Choquet Bruhat in occasion of her 80th year.