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.
| 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 | |
| Publication status | Published - 2006 |
| Externally published | Yes |
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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.
In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 462, No. 2071, 2006, p. 1949-1963.Research output: Contribution to journal › Article
}
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 -