Cauchy problem for a higher order generalized Boussinesq-type equation with hydrodynamical damped term

Abstract

We consider the existence and nonexistence of global weak solutions to the Cauchy problem for a higher order generalized Boussinesq-type equation with hydrodynamical damped term in -dimensional space. The existence of global weak solutions is proved under the assumptions that the nonlinear term is polynomial growth order, either , , the constant , or the initial data belongs to the potential well. Moreover, we establish two finite-time blow up results for any weak solutions with negative initial energy or nonnegative initial energy using the concavity method.