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

Research output: Contribution to journal › Article

11 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 Jan 1
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)

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Cho, Y., & Ozawa, T. (2006). Remarks on modified improved Boussinesq equations in one space dimension. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 462(2071), 1949-1963. https://doi.org/10.1098/rspa.2006.1675

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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.",

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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.

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