Remarks on modified improved Boussinesq equations in one space dimension

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