Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions revisited

Nakao Hayashi; Pavel I. Naumkin; Satoshi Tonegawa

doi:10.1007/s00033-011-0183-7

Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions revisited

Nakao Hayashi, Pavel I. Naumkin, Satoshi Tonegawa

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We continue to study the existence of the wave operators for the nonlinear Klein-Gordon equation with quadratic nonlinearity in two space dimensions (∂ _t ² - Δ + m ²) u = λu ², (t,x) ∈ R × R ². We prove that if, where γ ∈ (0, 1/4) and the norm, then there exist ρ > 0 and T > 1 such that the nonlinear Klein-Gordon equation has a unique global solution u ∈ C([T, ∞); H ^1/2 (R ²)) satisfying the asymptotics, for all t > T, where u ₀ denotes the solution of the free Klein-Gordon equation.

Original language	English
Pages (from-to)	655-673
Number of pages	19
Journal	Zeitschrift fur Angewandte Mathematik und Physik
Volume	63
Issue number	4
DOIs	https://doi.org/10.1007/s00033-011-0183-7
Publication status	Published - 2012 Aug
Externally published	Yes

Keywords

Nonlinear Klein-Gordon equations
Quadratic nonlinearity
Two space dimensions

ASJC Scopus subject areas

Mathematics(all)
Physics and Astronomy(all)
Applied Mathematics

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10.1007/s00033-011-0183-7

Cite this

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title = "Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions revisited",

abstract = "We continue to study the existence of the wave operators for the nonlinear Klein-Gordon equation with quadratic nonlinearity in two space dimensions (∂ t 2 - Δ + m 2) u = λu 2, (t,x) ∈ R × R 2. We prove that if, where γ ∈ (0, 1/4) and the norm, then there exist ρ > 0 and T > 1 such that the nonlinear Klein-Gordon equation has a unique global solution u ∈ C([T, ∞); H 1/2 (R 2)) satisfying the asymptotics, for all t > T, where u 0 denotes the solution of the free Klein-Gordon equation.",

keywords = "Nonlinear Klein-Gordon equations, Quadratic nonlinearity, Two space dimensions",

author = "Nakao Hayashi and Naumkin, {Pavel I.} and Satoshi Tonegawa",

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doi = "10.1007/s00033-011-0183-7",

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T1 - Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions revisited

AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

AU - Tonegawa, Satoshi

PY - 2012/8

Y1 - 2012/8

N2 - We continue to study the existence of the wave operators for the nonlinear Klein-Gordon equation with quadratic nonlinearity in two space dimensions (∂ t 2 - Δ + m 2) u = λu 2, (t,x) ∈ R × R 2. We prove that if, where γ ∈ (0, 1/4) and the norm, then there exist ρ > 0 and T > 1 such that the nonlinear Klein-Gordon equation has a unique global solution u ∈ C([T, ∞); H 1/2 (R 2)) satisfying the asymptotics, for all t > T, where u 0 denotes the solution of the free Klein-Gordon equation.

AB - We continue to study the existence of the wave operators for the nonlinear Klein-Gordon equation with quadratic nonlinearity in two space dimensions (∂ t 2 - Δ + m 2) u = λu 2, (t,x) ∈ R × R 2. We prove that if, where γ ∈ (0, 1/4) and the norm, then there exist ρ > 0 and T > 1 such that the nonlinear Klein-Gordon equation has a unique global solution u ∈ C([T, ∞); H 1/2 (R 2)) satisfying the asymptotics, for all t > T, where u 0 denotes the solution of the free Klein-Gordon equation.

KW - Nonlinear Klein-Gordon equations

KW - Quadratic nonlinearity

KW - Two space dimensions

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Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions revisited

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Keywords

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