The Cauchy Problem for Nonlinear Klein-Gordon Equations in the Sobolev Spaces

Makoto Nakamura, Tohru Ozawa

Research output: Contribution to journalArticle

36 Citations (Scopus)

Abstract

The local and global well-posedness for the Cauchy problem for a class of nonlinear Klein-Gordon equations is studied in the Sobolev space Hs = Hs(Rn) with s ≥ n/2. The global well-posedness of the problem is proved under the following assumptions: (1) Concerning the nonlinearity f, f(u) behaves as a power u1+4/n near zero. At infinity f(u) has an exponential growth rate such as exp(κ|u|ν) with κ > 0 and 0<ν≤2 if s = n/2, and has an arbitrary growth rate if s > n/2. (2) Concerning the Cauchy data (ϕ,Ψ) e Hs = Hs ⊕ HS-1, ‖(ϕ,Ψ); H1/2‖ is relatively small with respect to ‖(ϕ,Ψ); Hs∗‖, where s∗ is a number with s∗ = n/2 if s = n/2, n/2 < s∗ ≤ s if s > n/2, and the smallness of ‖(ϕ,Ψ); Hn/2‖ is also needed when s = n/2 and ν = 2.

Original languageEnglish
Pages (from-to)255-293
Number of pages39
JournalPublications of the Research Institute for Mathematical Sciences
Volume37
Issue number3
DOIs
Publication statusPublished - 2001
Externally publishedYes

Fingerprint

Nonlinear Klein-Gordon Equation
Global Well-posedness
Sobolev Spaces
Cauchy Problem
Exponential Growth
Cauchy
Infinity
Nonlinearity
Zero
Class

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

The Cauchy Problem for Nonlinear Klein-Gordon Equations in the Sobolev Spaces. / Nakamura, Makoto; Ozawa, Tohru.

In: Publications of the Research Institute for Mathematical Sciences, Vol. 37, No. 3, 2001, p. 255-293.

Research output: Contribution to journalArticle

@article{099b8e69f22f4c5aac9c5c6c73b2d98c,
title = "The Cauchy Problem for Nonlinear Klein-Gordon Equations in the Sobolev Spaces",
abstract = "The local and global well-posedness for the Cauchy problem for a class of nonlinear Klein-Gordon equations is studied in the Sobolev space Hs = Hs(Rn) with s ≥ n/2. The global well-posedness of the problem is proved under the following assumptions: (1) Concerning the nonlinearity f, f(u) behaves as a power u1+4/n near zero. At infinity f(u) has an exponential growth rate such as exp(κ|u|ν) with κ > 0 and 0<ν≤2 if s = n/2, and has an arbitrary growth rate if s > n/2. (2) Concerning the Cauchy data (ϕ,Ψ) e Hs = Hs ⊕ HS-1, ‖(ϕ,Ψ); H1/2‖ is relatively small with respect to ‖(ϕ,Ψ); Hs∗‖, where s∗ is a number with s∗ = n/2 if s = n/2, n/2 < s∗ ≤ s if s > n/2, and the smallness of ‖(ϕ,Ψ); Hn/2‖ is also needed when s = n/2 and ν = 2.",
author = "Makoto Nakamura and Tohru Ozawa",
year = "2001",
doi = "10.2977/prims/1145477225",
language = "English",
volume = "37",
pages = "255--293",
journal = "Publications of the Research Institute for Mathematical Sciences",
issn = "0034-5318",
publisher = "European Mathematical Society Publishing House",
number = "3",

}

TY - JOUR

T1 - The Cauchy Problem for Nonlinear Klein-Gordon Equations in the Sobolev Spaces

AU - Nakamura, Makoto

AU - Ozawa, Tohru

PY - 2001

Y1 - 2001

N2 - The local and global well-posedness for the Cauchy problem for a class of nonlinear Klein-Gordon equations is studied in the Sobolev space Hs = Hs(Rn) with s ≥ n/2. The global well-posedness of the problem is proved under the following assumptions: (1) Concerning the nonlinearity f, f(u) behaves as a power u1+4/n near zero. At infinity f(u) has an exponential growth rate such as exp(κ|u|ν) with κ > 0 and 0<ν≤2 if s = n/2, and has an arbitrary growth rate if s > n/2. (2) Concerning the Cauchy data (ϕ,Ψ) e Hs = Hs ⊕ HS-1, ‖(ϕ,Ψ); H1/2‖ is relatively small with respect to ‖(ϕ,Ψ); Hs∗‖, where s∗ is a number with s∗ = n/2 if s = n/2, n/2 < s∗ ≤ s if s > n/2, and the smallness of ‖(ϕ,Ψ); Hn/2‖ is also needed when s = n/2 and ν = 2.

AB - The local and global well-posedness for the Cauchy problem for a class of nonlinear Klein-Gordon equations is studied in the Sobolev space Hs = Hs(Rn) with s ≥ n/2. The global well-posedness of the problem is proved under the following assumptions: (1) Concerning the nonlinearity f, f(u) behaves as a power u1+4/n near zero. At infinity f(u) has an exponential growth rate such as exp(κ|u|ν) with κ > 0 and 0<ν≤2 if s = n/2, and has an arbitrary growth rate if s > n/2. (2) Concerning the Cauchy data (ϕ,Ψ) e Hs = Hs ⊕ HS-1, ‖(ϕ,Ψ); H1/2‖ is relatively small with respect to ‖(ϕ,Ψ); Hs∗‖, where s∗ is a number with s∗ = n/2 if s = n/2, n/2 < s∗ ≤ s if s > n/2, and the smallness of ‖(ϕ,Ψ); Hn/2‖ is also needed when s = n/2 and ν = 2.

UR - http://www.scopus.com/inward/record.url?scp=85013020039&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85013020039&partnerID=8YFLogxK

U2 - 10.2977/prims/1145477225

DO - 10.2977/prims/1145477225

M3 - Article

VL - 37

SP - 255

EP - 293

JO - Publications of the Research Institute for Mathematical Sciences

JF - Publications of the Research Institute for Mathematical Sciences

SN - 0034-5318

IS - 3

ER -