Remarks on modified improved Boussinesq equations in one space dimension

Yonggeun Cho, Tohru Ozawa

Research output: Contribution to journalArticle

11 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 Jan 1
Externally publishedYes


  • 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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