### 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 |
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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 | |

Publication status | Published - 2006 |

Externally published | Yes |

### Fingerprint

### Keywords

- Global existence
- Modified improved Boussinesq equation
- Scattering
- Small amplitude solution

### ASJC Scopus subject areas

- General

### Cite this

**Remarks on modified improved Boussinesq equations in one space dimension.** / Cho, Yonggeun; Ozawa, Tohru.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 462, no. 2071, pp. 1949-1963. https://doi.org/10.1098/rspa.2006.1675

}

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

VL - 462

SP - 1949

EP - 1963

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

IS - 2071

ER -