Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension

Frank Merle; Hatem Zaag

Displaying similar documents to “Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension”

Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle (2011)

Journal of the European Mathematical Society

Similarity:

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...

Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle (2012)

Journal of the European Mathematical Society

Similarity:

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.

The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes

Jonathan Luk (2013)

Journal of the European Mathematical Society

Similarity:

We study a semilinear equation with derivatives satisfying a null condition on slowly rotating Kerr spacetimes. We prove that given sufficiently small initial data, the solution exists globally in time and decays with a quantitative rate to the trivial solution. The proof uses the robust vector field method. It makes use of the decay properties of the linear wave equation on Kerr spacetime, in particular the improved decay rates in the region ${r \leq \frac{t}{4}}$ .

Stable blow up dynamics for the critical co-rotational Wave Maps and equivariant Yang-Mills Problems

Pierre Raphaël, Igor Rodnianski (2008-2009)

Séminaire Équations aux dérivées partielles

Similarity:

This note summarizes the results obtained in []. 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 $S O (4)$ Yang-Mills problem. We derive sharp asymptotics on the dynamics at blow up time and prove quantization of the energy focused at the singularity.

Blow up and near soliton dynamics for the $L^{2}$ critical gKdV equation

Yvan Martel, Frank Merle, Pierre Raphaël (2011-2012)

Séminaire Laurent Schwartz — EDP et applications

Similarity:

These notes present the main results of [, , ] concerning the mass critical (gKdV) equation $u_{t} + {(u_{x x} + u^{5})}_{x} = 0$ for initial data in $H^{1}$ close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in $H^{1}$ , construction of various exotic blow up rates in $H^{1}$ , including grow up in infinite time. ...

Recent progress in attractors for quintic wave equations

Anton Savostianov, Sergey Zelik (2014)

Mathematica Bohemica

Similarity:

We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of $ℝ^{3}$ with damping terms of the form ${(- Δ_{x})}^{θ} \partial_{t} u$ , where $θ = 0$ or $θ = 1 / 2$ . The main ingredient of the work is the hidden extra regularity of solutions that does not follow from energy estimates. Due to the extra regularity of solutions existence of a smooth attractor then follows from the smoothing property when $θ = 1 / 2$ . For $θ = 0$ existence of smooth attractors is more complicated...

Blow-Up of Solutions of Nonlinear Wave Equations in Three Space Dimensions.

Fritz John (1979)

Manuscripta mathematica

Similarity:

The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation

Juan Luis Vázquez (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

We review the main mathematical questions posed in blow-up problems for reaction-diffusion equations and discuss results of the author and collaborators on the subjects of continuation of solutions after blow-up, existence of transient blow-up solutions (so-called peaking solutions) and avalanche formation as a mechanism of complete blow-up.