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 $$M^{s}_{p^\prime ,q}(\mathbb {R}^n),$$ $$n\ge 1.$$ 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 $$ M^s_{p,q}\times D^{-1}JM^s_{p,q}$$ -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.