Existence theory for the Boussinesq equation in Modulation spaces

Abstract

In this paper we study the Cauchy problem for the generalized Boussinesq equation with initial data in modulation spaces $M_{p^{\prime },q}^s({\mathbb{R}}^n),$ $n\ge 1.$ After a decomposition of the Boussinesq equation in a $2\times 2$-nonlinear system, we obtain the existence of global and local solutions in several classes of functions with values in $M_{p,q}^s\times D^{-1}J{M}_{p,q}^s$ spaces for suitable $p,q$ and $s,$ including the special case $p=2,q=1$ and $s=0.$ Finally, we prove some results of scattering and asymptotic stability in the framework of modulation spaces.

19 pages