Remarks on modified improved Boussinesq equations in one space dimension

Yonggeun Cho; Tohru Ozawa

doi:10.1098/rspa.2006.1675

Remarks on modified improved Boussinesq equations in one space dimension

Yonggeun Cho^*, Tohru Ozawa

^*Corresponding author for this work

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

13 Citations (Scopus)

Abstract

We study the existence and scattering of global small amplitude solutions to modified improved Boussinesq equations in one dimension with nonlinear term f(u) behaving as a power u^p as u → 0. Solutions are considered in H^s space for all s> 1/2. According to the value of s, the power nonlinearity exponent p is determined. Liu (Liu 1996 Indiana Univ. Math. J. 45, 797-816) obtained the minimum value of p greater than 8 at s = 3/2 for sufficiently small Cauchy data. In this paper, we prove that p can be reduced to be greater than 9/2 at s> 17/10 and the corresponding solution u has the time decay, such as ∥u(t)∥_L∞ = O(t^-2/5) as t → ∞. We also prove non-existence of non-trivial asymptotically free solutions for 1 < p ≤ 2 under vanishing condition near zero frequency on asymptotic states.

Original language	English
Pages (from-to)	1949-1963
Number of pages	15
Journal	Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume	462
Issue number	2071
DOIs	https://doi.org/10.1098/rspa.2006.1675
Publication status	Published - 2006
Externally published	Yes

Keywords

Global existence
Modified improved Boussinesq equation
Scattering
Small amplitude solution

ASJC Scopus subject areas

Mathematics(all)
Engineering(all)
Physics and Astronomy(all)

Access to Document

10.1098/rspa.2006.1675

Cite this

@article{6ed4977a31054f21872db939655bd01d,

title = "Remarks on modified improved Boussinesq equations in one space dimension",

abstract = "We study the existence and scattering of global small amplitude solutions to modified improved Boussinesq equations in one dimension with nonlinear term f(u) behaving as a power up as u → 0. Solutions are considered in Hs space for all s> 1/2. According to the value of s, the power nonlinearity exponent p is determined. Liu (Liu 1996 Indiana Univ. Math. J. 45, 797-816) obtained the minimum value of p greater than 8 at s = 3/2 for sufficiently small Cauchy data. In this paper, we prove that p can be reduced to be greater than 9/2 at s> 17/10 and the corresponding solution u has the time decay, such as ∥u(t)∥L∞ = O(t-2/5) as t → ∞. We also prove non-existence of non-trivial asymptotically free solutions for 1 < p ≤ 2 under vanishing condition near zero frequency on asymptotic states.",

keywords = "Global existence, Modified improved Boussinesq equation, Scattering, Small amplitude solution",

author = "Yonggeun Cho and Tohru Ozawa",

year = "2006",

doi = "10.1098/rspa.2006.1675",

language = "English",

volume = "462",

pages = "1949--1963",

journal = "PROC. R. SOC. - A.",

issn = "0950-1207",

publisher = "Royal Society of London",

number = "2071",

}

TY - JOUR

T1 - Remarks on modified improved Boussinesq equations in one space dimension

AU - Cho, Yonggeun

AU - Ozawa, Tohru

PY - 2006

Y1 - 2006

N2 - We study the existence and scattering of global small amplitude solutions to modified improved Boussinesq equations in one dimension with nonlinear term f(u) behaving as a power up as u → 0. Solutions are considered in Hs space for all s> 1/2. According to the value of s, the power nonlinearity exponent p is determined. Liu (Liu 1996 Indiana Univ. Math. J. 45, 797-816) obtained the minimum value of p greater than 8 at s = 3/2 for sufficiently small Cauchy data. In this paper, we prove that p can be reduced to be greater than 9/2 at s> 17/10 and the corresponding solution u has the time decay, such as ∥u(t)∥L∞ = O(t-2/5) as t → ∞. We also prove non-existence of non-trivial asymptotically free solutions for 1 < p ≤ 2 under vanishing condition near zero frequency on asymptotic states.

AB - We study the existence and scattering of global small amplitude solutions to modified improved Boussinesq equations in one dimension with nonlinear term f(u) behaving as a power up as u → 0. Solutions are considered in Hs space for all s> 1/2. According to the value of s, the power nonlinearity exponent p is determined. Liu (Liu 1996 Indiana Univ. Math. J. 45, 797-816) obtained the minimum value of p greater than 8 at s = 3/2 for sufficiently small Cauchy data. In this paper, we prove that p can be reduced to be greater than 9/2 at s> 17/10 and the corresponding solution u has the time decay, such as ∥u(t)∥L∞ = O(t-2/5) as t → ∞. We also prove non-existence of non-trivial asymptotically free solutions for 1 < p ≤ 2 under vanishing condition near zero frequency on asymptotic states.

KW - Global existence

KW - Modified improved Boussinesq equation

KW - Scattering

KW - Small amplitude solution

UR - http://www.scopus.com/inward/record.url?scp=33845538891&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33845538891&partnerID=8YFLogxK

U2 - 10.1098/rspa.2006.1675

DO - 10.1098/rspa.2006.1675

M3 - Article

AN - SCOPUS:33845538891

VL - 462

SP - 1949

EP - 1963

JO - PROC. R. SOC. - A.

JF - PROC. R. SOC. - A.

SN - 0950-1207

IS - 2071

ER -

Remarks on modified improved Boussinesq equations in one space dimension

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this