Initial Boundary Value Problem For The Double Dispersion Equation

In this paper we consider the initial boundary value Problem of the double dispersion equation. We prove the existence and uniqueness of local solutions by Faedo-Galerkin method. When β> 0, we give the proof of the well-posedness of global solutions. Besides, when β< 0, under three different cases, namely, E(0)< d, E(0)= d and E(0)> d, we prove the existence of the global solutions in energy space by potential well method and provide sufficient conditions of finite time blow up solutions. The asymptotic behavior of the global solutions is also studied.