# Remarks on modified improved Boussinesq equations in one space dimension

Preprint Series # 723
Cho, Yonggeun and Ozawa, Tohru Remarks on modified improved Boussinesq equations in one space dimension. (2005);

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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 $u^p$ as $u \to 0$. Solutions in $H^s$ space are considered for all $s > 0$. According to the value of $s$, the power nonlinearity exponent $p$ is determined. Liu \cite{liu} obtained the minimum value of $p$ greater than $8$ at $s = \frac32$ for sufficiently small Cauchy data. In this paper, we prove that $p$ can be reduced to be greater than $\frac92$ at $s > \frac85$ and the corresponding solution $u$ has the time decay such as $\|u( t)\|_{L^\infty} = O(t^{-\frac25})$ as $t \to \infty$. We also prove nonexistence of nontrivial asymptotically free solutions for $1 < p \le 2$ under vanishing condition near zero frequency on asymptotic states.

Item Type: Preprint 20 IMBq equation, small amplitude solution, global existence, scattering 35-xx PARTIAL DIFFERENTIAL EQUATIONS 969