Related papers: Slow blow-up solutions for the H^1(R^3) critical f…

In this paper we study the Euler-Poincar\'{e} equations in $\Bbb R^N$. We prove local existence of weak solutions in $W^{2,p}(\Bbb R^N),$ $p>N$, and local existence of unique classical solutions in $H^k (\Bbb R^N)$, $k>N/2+3$, as well as a…

We fully revisit the near soliton dynamics for the mass critical (gKdV) equation. In Part I, for a class of initial data close to the soliton, we prove that only three scenario can occur: (BLOW UP) the solution blows up in finite time $T$…

We consider the half-wave equation $i u_t=Du-|u|^{\frac{2}{3}}u$ in three dimension and in the mass critical. For initial data $u(t_0,x)=u_0(x)\in H^{1/2}_{rad}(\mathbb{R}^3)$ with radial symmetry, we construct a new class of minimal mass…

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.

In this work we consider an energy subcritical semi-linear wave equation ($3 < p < 5$) \[ \partial_t^2 u - \Delta u = \phi(x) |u|^{p-1} u, \qquad (x,t) \in {\mathbb R}^3 \times {\mathbb R} \] with initial data $(u,u_t)|_{t=0} = (u_0,u_1)\in…

We construct multisoliton solutions to the defocusing energy critical wave equation with potentials in $\mathbb{R}^{3}$ based on regular and reversed Strichartz estimates developed in \cite{GC3} for wave equations with charge transfer…

The focusing critical wave equation in three dimensions exhibits a special class of static solutions which are linearly unstable. These solutions decay like an inverse first power. We construct small codimension one stable manifolds in the…

We study the blowup behavior for the focusing energy-supercritical semilinear wave equation in 3 space dimensions without symmetry assumptions on the data. We prove the stability of the ODE blowup profile.

This paper is concerned with the energy decay and the finite time blow-up of the solution to a viscoelastic wave equation with polynomial nonlinearity and weak damping. We establish explicit and general decay results for the solutions by…

We consider the nonlinear Schr\"odinger equation \[ u_t = i \Delta u + | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) \alpha \le 4$. Under the additional technical…

We study the energy critical wave equation in 3 dimensions around a single soliton. We obtain energy boundedness (modulo unstable modes) for the linearised problem. We use this to construct scattering solutions in a neighbourhood of…

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…

In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power nonlinearity in one space dimension. We first characterize all the solutions of the associated stationary problem as a two-parameter…

The paper is concerned with the problem of explosive solutions for a class of nonlinear stochastic wave equations in a domain $\mathcal{D}\subset\mathbb{R}^d$ for $d\leq3$. Under appropriate conditions on the initial data, the nonlinear…

This paper deals with blow-up for the complex-valued semilinear wave equation with power nonlinearity in dimension 1. Up to a rotation of the solution in the complex plane, we show that near a characteristic blow-up point, the solution…

We consider energy-critical damped wave equation \begin{equation*} \partial_{tt}u-\Delta u+\alpha \partial_t u=\left|u\right|^{\frac{4}{D-2}}u \end{equation*} with radial initial data in dimensions $D\geq 4$. The equation has a nontrivial…

We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation \[ \Box u = -u^5 \] on $\mathbb R^{3+1}$ constructed by Krieger-Schlag-Tataru are stable along a co-dimension one Lipschitz manifold of…

For the quintic, mass critical generalized Korteweg-de Vries equation, for any $\nu \in (\frac{1}{2}, 1)$, we prove the existence of solutions in the energy space that blow up in finite time $T>0$ with the blow-up rate $\|\partial_x…

This paper is devoted to study a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under suitable conditions, we prove that any weak solutions with negative initial energy…

Consider the energy-critical focusing wave equation in odd space dimension $N\geq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. In this paper we prove that any radial, bounded…