Potential well method for initial boundary value problem of the generalized double dispersion equations

Abstract

In this paper we study the initial boundary value problem of thegeneralized double dispersion equations$u_{t t}-u_{x x}-u_{x x t t}+u_{x x x x}=f(u)_{x x}$, where $f(u)$ includeconvex function as a special case. By introducing a family ofpotential wells we first prove the invariance of some sets andvacuum isolating of solutions, then we obtain a threshold result ofglobal existence and nonexistence of solutions. Finally we discussthe global existence of solutions for problem with critical initialcondition.