The defocusing energy-critical Klein-Gordon-Hartree equation

Qianyun Miao; Jiqiang Zheng

Displaying similar documents to “The defocusing energy-critical Klein-Gordon-Hartree equation”

Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case

Changxing Miao, Guixiang Xu, Lifeng Zhao (2009)

Colloquium Mathematicae

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We establish global existence and scattering for radial solutions to the energy-critical focusing Hartree equation with energy and Ḣ¹ norm less than those of the ground state in $ℝ \times ℝ^{d}$ , d ≥ 5.

Dynamics of a modified Davey-Stewartson system in ℝ³

Jing Lu (2016)

Colloquium Mathematicae

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We study the Cauchy problem in ℝ³ for the modified Davey-Stewartson system $i \partial ₜ u + Δ u = λ ₁ | u | ⁴ u + λ ₂ b ₁ u v_{x ₁}$ , $- Δ v = b ₂ {(| u | ²)}_{x ₁}$ . Under certain conditions on λ₁ and λ₂, we provide a complete picture of the local and global well-posedness, scattering and blow-up of the solutions in the energy space. Methods used in the paper are based upon the perturbation theory from [Tao et al., Comm. Partial Differential Equations 32 (2007), 1281-1343] and the convexity method from [Glassey, J. Math. Phys. 18 (1977), 1794-1797].

On the real analyticity of the scattering operator for the Hartree equation

Changxing Miao, Haigen Wu, Junyong Zhang (2009)

Annales Polonici Mathematici

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We study the real analyticity of the scattering operator for the Hartree equation $i \partial_{t} u = - Δ u + u (V * | u | ²)$ . To this end, we exploit interior and exterior cut-off in time and space, together with a compactness argument to overcome difficulties which arise from absence of good properties as for the Klein-Gordon equation, such as the finite speed of propagation and ideal time decay estimate. Additionally, the method in this paper allows us to simplify the proof of analyticity of the scattering operator for the...

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.

Scattering for 1D cubic NLS and singular vortex dynamics

Valeria Banica, Luis Vega (2012)

Journal of the European Mathematical Society

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We study the stability of self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions $χ_{a} (t, x)$ form a family of evolving regular curves in $ℝ^{3}$ that develop a singularity in finite time, indexed by a parameter $a > 0$ . We consider curves that are small regular perturbations of $χ_{a} (t_{0}, x)$ for a fixed time $t_{0}$ . In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of...

Absolutely continuous and singular spectral shift functions

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Given a self-adjoint operator H₀, a self-adjoint trace-class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and cokernel, using the limiting absorption principle an explicit set Λ(H₀;F) ⊂ ℝ of full Lebesgue measure is defined, such that for all λ ∈ Λ(H₀+rV;F) ∩ Λ(H₀;F), where r ∈ ℝ, the wave $w_{\pm} (λ; H ₀ + r V, H ₀)$ and the scattering matrices S(λ;H₀+rV,H₀) can be defined unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including...

The $n$ -centre problem of celestial mechanics for large energies

Andreas Knauf (2002)

Journal of the European Mathematical Society

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We consider the classical three-dimensional motion in a potential which is the sum of $n$ attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the $n$ centres, we find a universal behaviour for all energies $E$ above a positive threshold. Whereas for $n = 1$ there are no bounded orbits, and for $n = 2$ there is just one closed orbit, for $n \geq 3$ the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of...

Global existence and stability of solution for a nonlinear Kirchhoff type reaction-diffusion equation with variable exponents

Aya Khaldi, Amar Ouaoua, Messaoud Maouni (2022)

Mathematica Bohemica

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We consider a class of Kirchhoff type reaction-diffusion equations with variable exponents and source terms $u_{t} - M (\int_{Ω} {| \nabla u |}^{2} d x) Δ u + {| u |}^{m (x) - 2} u_{t} = {| u |}^{r (x) - 2} u .$ We prove with suitable assumptions on the variable exponents $r (\cdot),$ $m (\cdot)$ the global existence of the solution and a stability result using potential and Nihari’s functionals with small positive initial energy, the stability being based on Komornik’s inequality.

Convergence of minimax structures and continuation of critical points for singularly perturbed systems

Benedetta Noris, Hugo Tavares, Susanna Terracini, Gianmaria Verzini (2012)

Journal of the European Mathematical Society

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In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system $- Δ u + u^{3} + β u v^{2} = λ u, - Δ v + v^{3} + β u^{2} v = μ v, u, v \in H_{0}^{1} (Ω), u, v > 0$ , as the interspecies scattering length $β$ goes to $+ \infty$ . For this system we consider the associated energy functionals $J_{β}, β \in (0, + \infty)$ , with $L^{2}$ -mass constraints, which limit $J_{\infty}$ (as $β \to + \infty$ ) is strongly irregular. For such functionals, we construct multiple critical points...

Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent

Lan Zeng, Chun Lei Tang (2016)

Annales Polonici Mathematici

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We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ $- [a + b (\int_{Ω} {| \nabla u | ² d x)}^{m} {] Δ u = f (x, u) + | u |}^{2 * - 2} u$ in Ω, ⎨ ⎩ u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.

Radiation conditions at the top of a rotational cusp in the theory of water-waves

Sergey A. Nazarov, Jari Taskinen (2011)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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We study the linearized water-wave problem in a bounded domain (a finite pond of water) of $ℝ^{3}$ , having a cuspidal boundary irregularity created by a submerged body. In earlier publications the authors discovered that in this situation the spectrum of the problem may contain a continuous component in spite of the boundedness of the domain. Here, we proceed to impose and study radiation conditions at a point $𝒪$ of the water surface, where a submerged body touches the surface (see Fig. 1)....

On the distribution of scattering poles for perturbations of the Laplacian

Georgi Vodev (1992)

Annales de l'institut Fourier

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We consider selfadjoint positively definite operators of the form $- Δ + P$ (not necessarily elliptic) in $ℝ^{n}$ , $n \geq 3$ , odd, where $P$ is a second-order differential operator with coefficients of compact supports. We show that the number of the scattering poles outside a conic neighbourhood of the real axis admits the same estimates as in the elliptic case. More precisely, if ${λ_{j}} (Im λ_{j} \geq 0_{)}$ are the scattering poles associated to the operator $- Δ + P$ repeated according to multiplicity, it is proved that for any $ϵ > 0$ there exists...

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

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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.

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...

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

Philip Brenner (2003)

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Local energy decay for several evolution equations on asymptotically euclidean manifolds

Jean-François Bony, Dietrich Häfner (2012)

Annales scientifiques de l'École Normale Supérieure

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Let $P$ be a long range metric perturbation of the Euclidean Laplacian on $ℝ^{d}$ , $d \geq 2$ . We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to $P$ . The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group $e^{i t f (P)}$ where $f$ has a suitable development at zero (resp. infinity). ...

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,...

About global existence and asymptotic behavior for two dimensional gravity water waves

Thomas Alazard (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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The main result of this talk is a global existence theorem for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds. The proof is based on a bootstrap argument involving $L^{2}$ and $L^{\infty}$ estimates. The $L^{2}$ bounds are proved in the paper [5]. They rely on a normal forms paradifferential method allowing one to obtain energy estimates...

Existence and multiplicity results for a nonlinear stationary Schrödinger equation

Danila Sandra Moschetto (2010)

Annales Polonici Mathematici

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We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), $x \in ℝ^{N}$ , $u \in H ¹ (ℝ^{N})$ , where λ is a positive parameter, a and b are positive functions, while $f : ℝ \to ℝ$ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

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

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In this paper we consider a nonlinear elliptic equation with critical growth for the operator $\Delta^{2}$ in a bounded domain $\Omega\subset\mathbb{R}^{n}$ . We state some existence results when $n\geq 8$ . Moreover, we consider $5\leq n\leq 7$ , expecially when $\Omega$ is a ball in $\mathbb{R}^{n}$ .

Critical points via $Γ$ -convergence: general theory and applications

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Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential

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Journal of the European Mathematical Society

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We consider a singularly perturbed elliptic equation $ϵ^{2} Δ u - V (x) u + f (u) = 0, u (x) > 0$ on $ℝ^{N}$ , ${𝚕𝚒𝚖}_{|x| \to \infty} u (x) = 0$ , where $V (x) > 0$ for any $x \in ℝ^{N}$ . The singularly perturbed problem has corresponding limiting problems $Δ U - c U + f (U) = 0, U (x) > 0$ on $ℝ^{N}$ , ${𝚕𝚒𝚖}_{|x| \to \infty} U (x) = 0, c > 0$ . Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity $f$ for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential $V$ under possibly general conditions...

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Colloquium Mathematicae

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We consider the Neumann problem for the equation $- Δ u - λ u = {Q (x) | u |}^{2 * - 2} u$ , u ∈ H¹(Ω), where Q is a positive and continuous coefficient on Ω̅ and λ is a parameter between two consecutive eigenvalues $λ_{k - 1}$ and $λ_{k}$ . Applying a min-max principle based on topological linking we prove the existence of a solution.

Blow-up of the solution to the initial-value problem in nonlinear three-dimensional hyperelasticity

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We consider the initial value problem for the nonlinear partial differential equations describing the motion of an inhomogeneous and anisotropic hyperelastic medium. We assume that the stored energy function of the hyperelastic material is a function of the point x and the nonlinear Green-St. Venant strain tensor $e_{j k}$ . Moreover, we assume that the stored energy function is $C^{\infty}$ with respect to x and $e_{j k}$ . In our description we assume that Piola-Kirchhoff’s stress tensor $p_{j k}$ depends on the tensor...

Invariants, conservation laws and time decay for a nonlinear system of Klein-Gordon equations with Hamiltonian structure

Changxing Miao, Youbin Zhu (2006)

Applicationes Mathematicae

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We discuss invariants and conservation laws for a nonlinear system of Klein-Gordon equations with Hamiltonian structure ⎧ $u_{t t} - Δ u + m ² u = - F ₁ (| u | ², | v | ²) u$ , ⎨ ⎩ $v_{t t} - Δ v + m ² v = - F ₂ (| u | ², | v | ²) v$ for which there exists a function F(λ,μ) such that ∂F(λ,μ)/∂λ = F₁(λ,μ), ∂F(λ,μ)/∂μ = F₂(λ,μ). Based on Morawetz-type identity, we prove that solutions to the above system decay to zero in local L²-norm, and local energy also decays to zero if the initial energy satisfies $E (u, v, ℝ ⁿ, 0) = 1 / 2 \int_{ℝ ⁿ} (| \nabla u (0) | ² + | u_{t} (0) | ² + m ² | u (0) | ² + | \nabla v (0) | ² + | v_{t} (0) | ² + m ² | v (0) | ² + F (| u (0) | ², | v (0) | ²)) d x < \infty$ , and F₁(|u|²,|v|²)|u|² + F₂(|u|²,|v|²)|v|² - F(|u|²,|v|²) ≥ aF(|u|²,|v|²) ≥ 0, a >...