Global existence and blow-up for the generalized sixth-order Boussinesq equation

In this paper we prove local well-posedness in L²(R)$L^2\left(\mathbb{R}\right)$ and H¹(R)$H^1\left(\mathbb{R}\right)$ for the generalized sixth-order Boussinesq equation _utt=_uxx+β_uxxxx+_uxxxxxx+_(|u|αu)xx$u_{tt}=u_{xx}+\beta u_{xxxx}+u_{xxxxxx}+{\left({\left|u\right|}^{\alpha }u\right)}_{xx}$. Our proof relies in the oscillatory integrals estimates introduced by Kenig et al. (1991) 14]. We also show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem.