Remarks on modified improved Boussinesq equations in one space dimension

Yonggeun Cho*, Tohru Ozawa

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)


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 languageEnglish
Pages (from-to)1949-1963
Number of pages15
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2071
Publication statusPublished - 2006
Externally publishedYes


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

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering
  • General Physics and Astronomy


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