This note summarizes the results obtained in [30]. We exhibit stable finite time blow up regimes for the energy critical co-rotational Wave Map with the ${𝕊}^2$ target in all homotopy classes and for the equivariant critical $SO\left(4\right)$ Yang-Mills problem. We derive sharp asymptotics on the dynamics at blow up time and prove quantization of the energy focused at the singularity.

This paper deals with blow-up properties of solutions to a semilinear parabolic system with weighted localized terms, subject to the homogeneous Dirichlet boundary conditions. We investigate the influence of the three factors: localized sources $u^p(x₀,t)$, vⁿ(x₀,t), local sources $u^m(x,t)$, $v^q(x,t)$, and weight functions a(x),b(x), on the asymptotic behavior of solutions. We obtain the uniform blow-up profiles not only for the cases m,q ≤ 1 or m,q > 1, but also for m > 1 q < 1 or m < 1 q > 1.

The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space ${\dot{B}}_{p,1}^{n/p-1}\left({ℝ}^n\right)\times {\dot{B}}_{p,1}^{n/p}\left({ℝ}^n\right)$ with $n<p<2n$ is established by using the heat semigroup theory and the Littlewood-Paley theory. The global well-posedness for the system is obtained with small initial datum by using the fixed point theorem. The blow-up results for strong solutions to the system are also analysed.

We consider solutions of quasilinear equations $u_t=\Delta u^m+u^p$ in ${ℝ}^N$ with the initial data $u_0$ satisfying $0<u_0<M$ and ${lim}_{\left|x\right|\to \infty }{u}_0\left(x\right)=M$ for some constant $M>0$. It is known that if $0<m<p$ with $p>1$, the blow-up set is empty. We find solutions $u$ that blow up throughout ${ℝ}^N$ when $m>p>1$.

We consider the focusing nonlinear Schrödinger equations $i{\partial }_{t}u+\Delta u+u{\left|u\right|}^{p-1}=0$. We prove the existence of two finite time blow up dynamics in the supercritical case and provide for each a qualitative description of the singularity formation near the blow up time.

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...

Following our previous paper in the radial case, we consider type II blow-up solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the equation, under an appropriate smallness assumption, any type II blow-up solution is asymptotically a regular solution plus a rescaled Lorentz transform of $W$ concentrating at the origin.