Potential well method for Cauchy problem of generalized double dispersion equations

Abstract

In this paper we study Cauchy problem of generalized 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 ) = | u | p , p > 1 or u 2 k , k = 1 , 2 , … . By introducing a family of potential wells we not only get a threshold result of global existence and nonexistence of solutions, but also obtain the invariance of some sets and vacuum isolating of solutions. In addition, the global existence and finite time blow up of solutions for problem with critical initial conditions E ( 0 ) = d , I ( u 0 ) ⩾ 0 or I ( u 0 ) < 0 are proved.