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

Displaying similar documents to “Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation”

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

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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.

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

Frank Merle, Hatem Zaag (2009-2010)

Séminaire Équations aux dérivées partielles

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We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution $u (x, t)$ , the graph $x \mapsto T (x)$ of its blow-up points and $𝒮 \subset ℝ$ the set of all characteristic points and show that $𝒮$ is locally finite. Finally, given $x_{0} \in 𝒮$ , we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons, with alternate signs and that...

Recent progress in attractors for quintic wave equations

Anton Savostianov, Sergey Zelik (2014)

Mathematica Bohemica

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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...

On the $L_{p}$ -decay and local energy decay of solutions to nonlinear Klein-Gordon equations

Philip Brenner (2003)

Banach Center Publications

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Blow-up of solutions for the Kirchhoff equation of q-Laplacian type with nonlinear dissipation

Abbes Benaissa, Salim A. Messaoudi (2002)

Colloquium Mathematicae

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We establish the blow-up of solutions to the Kirchhoff equation of q-Laplacian type with a nonlinear dissipative term $(| u_{t} |^{l - 2} u_{t})_{t} - M (| | A^{1 / 2} u | | ² ₂) A u + | u_{t} |^{β} u_{t} = {| u |}^{p} u$ , x ∈ Ω, t > 0.

Blow-up of solutions for the non-Newtonian polytropic filtration equation with a generalized source

Jun Zhou (2016)

Annales Polonici Mathematici

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This paper deals with the blow-up properties of the non-Newtonian polytropic filtration equation $u_{t} - d i v (| \nabla u^{m} |^{p - 2} \nabla u^{m}) = f (u)$ with homogeneous Dirichlet boundary conditions. The blow-up conditions, upper and lower bounds of the blow-up time, and the blow-up rate are established by using the energy method and differential inequality techniques.

Blow up for the critical gKdV equation. II: Minimal mass dynamics

Yvan Martel, Frank Merle, Pierre Raphaël (2015)

Journal of the European Mathematical Society

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We consider the mass critical (gKdV) equation $u_{t} + {(u_{x x} + u^{5})}_{x} = 0$ for initial data in $H^{1}$ . We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].

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

Jonathan Luk (2013)

Journal of the European Mathematical Society

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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 ${r \leq \frac{t}{4}}$ .

Focusing of a pulse with arbitrary phase shift for a nonlinear wave equation

Rémi Carles, David Lannes (2003)

Bulletin de la Société Mathématique de France

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We consider a system of two linear conservative wave equations, with a nonlinear coupling, in space dimension three. Spherical pulse like initial data cause focusing at the origin in the limit of short wavelength. Because the equations are conservative, the caustic crossing is not trivial, and we analyze it for particular initial data. It turns out that the phase shift between the incoming wave (before the focus) and the outgoing wave (past the focus) behaves like $ln ε$ , where $ε$ stands for...

Single-point blow-up for a semilinear parabolic system

Ph. Souplet (2009)

Journal of the European Mathematical Society

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We consider positive solutions of the system $u_{t} - Δ u = v^{p}$ ; $v_{t} - Δ v = u^{q}$ in a ball or in the whole space, with $p, q > 1$ . Relatively little is known on the blow-up set for semilinear parabolic systems and, up to now, no result was available for this basic system except for the very special case $p = q$ . Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for...

On blow-up for the Hartree equation

Jiqiang Zheng (2012)

Colloquium Mathematicae

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We study the blow-up of solutions to the focusing Hartree equation $i u_{t} + Δ u + {(| x |}^{- γ} * | u | ²) u = 0$ . We use the strategy derived from the almost finite speed of propagation ideas devised by Bourgain (1999) and virial analysis to deduce that the solution with negative energy (E(u₀) < 0) blows up in either finite or infinite time. We also show a result similar to one of Holmer and Roudenko (2010) for the Schrödinger equations using techniques from scattering theory.

The analysis of blow-up solutions to a semilinear parabolic system with weighted localized terms

Haihua Lu, Feng Wang, Qiaoyun Jiang (2011)

Annales Polonici Mathematici

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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 $u^{p} (x ₀, t)$ , 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 >...

The Cauchy problem for the liquid crystals system in the critical Besov space with negative index

Sen Ming, Han Yang, Zili Chen, Ls Yong (2017)

Czechoslovak Mathematical Journal

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The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space ${\dot{B}}_{p, 1}^{n / p - 1} (ℝ^{n}) \times {\dot{B}}_{p, 1}^{n / p} (ℝ^{n})$ with $n < p < 2 n$ 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.

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

Thierry Gallay, Romain Joly (2009)

Annales scientifiques de l'École Normale Supérieure

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We consider the damped wave equation $α u_{t t} + u_{t} = u_{x x} - V^{'} (u)$ on the whole real line, where $V$ is a bistable potential. This equation has travelling front solutions of the form $u (x, t) = h (x - s t)$ which describe a moving interface between two different steady states of the system, one of which being the global minimum of $V$ . We show that, if the initial data are sufficiently close to the profile of a front for large $| x |$ , the solution of the damped wave equation converges uniformly on $ℝ$ to a travelling front as $t \to + \infty$ . The proof of this...

Front propagation for nonlinear diffusion equations on the hyperbolic space

Hiroshi Matano, Fabio Punzo, Alberto Tesei (2015)

Journal of the European Mathematical Society

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We study the Cauchy problem in the hyperbolic space $ℍ^{n} (n \geq 2)$ for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space $ℝ^{n}$ new phenomena arise, which depend on the properties of the diffusion process in $ℍ^{n}$ . We also investigate a family of travelling wave solutions,...

Free decay of solutions to wave equations on a curved background

Serge Alinhac (2005)

Bulletin de la Société Mathématique de France

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We investigate for which metric $g$ (close to the standard metric $g_{0}$ ) the solutions of the corresponding d’Alembertian behave like free solutions of the standard wave equation. We give rather weak (, non integrable) decay conditions on $g - g_{0}$ ; in particular, $g - g_{0}$ decays like $t^{- \frac{1}{2} - ε}$ along wave cones.

Some decay properties for the damped wave equation on the torus

Nalini Anantharaman, Matthieu Léautaud (2012)

Journées Équations aux dérivées partielles

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This article is a proceedings version of the ongoing work [1], and has been the object of a talk of the second author during the Journées “Équations aux Dérivées Partielles” (Biarritz, 2012). We address the decay rates of the energy of the damped wave equation when the damping coefficient $b$ does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger...

A sharp Strichartz estimate for the wave equation with data in the energy space

Neal Bez, Keith M. Rogers (2013)

Journal of the European Mathematical Society

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We prove a sharp bilinear estimate for the wave equation from which we obtain the sharp constant in the Strichartz estimate which controls the $L_{t, x}^{4} (ℝ^{5 + 1})$ norm of the solution in terms of the energy. We also characterise the maximisers.

Dispersive and Strichartz estimates on H-type groups

Martin Del Hierro (2005)

Studia Mathematica

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Our purpose is to generalize the dispersive inequalities for the wave equation on the Heisenberg group, obtained in [1], to H-type groups. On those groups we get optimal time decay for solutions to the wave equation (decay as $t^{- p / 2}$ ) and the Schrödinger equation (decay as $t^{(1 - p) / 2}$ ), p being the dimension of the center of the group. As a corollary, we obtain the corresponding Strichartz inequalities for the wave equation, and, assuming that p > 1, for the Schrödinger equation.

Stability in exponential time of Minkowski space-time with a space-like translation symmetry

Cécile Huneau (2014-2015)

Séminaire Laurent Schwartz — EDP et applications

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In this note, we discuss the nonlinear stability in exponential time of Minkowski space-time with a translation space-like Killing field, proved in [13]. In the presence of such a symmetry, the $3 + 1$ vacuum Einstein equations reduce to the $2 + 1$ Einstein equations with a scalar field. We work in generalized wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in [13] is due to the decay in $1 / \sqrt{t}$ of free solutions...

Selfsimilar profiles in large time asymptotics of solutions to damped wave equations

Grzegorz Karch (2000)

Studia Mathematica

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Large time behavior of solutions to the generalized damped wave equation $u_{t t} + A u_{t} + ν B u + F (x, t, u, u_{t}, \nabla u) = 0$ for $(x, t) \in ℝ^{n} \times [0, \infty)$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where $A u_{t} = u_{t}$ , $B u = - Δ u$ , and the nonlinear term is either $| u_{t} |^{q - 1} u_{t}$ or ${| u |}^{α - 1} u$ . In this case, the asymptotic profile of solutions...