Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation

We study the Cauchy problem of generalized Boussinesq equation u t t − u x x + ( u x x + f ( u ) ) x x = 0 , where f ( u ) = ± | u | p or ± | u | p − 1 u , p > 1 . By introducing a family of potential wells we obtain invariant sets, vacuum isolating and threshold result of global existence and nonexistence of solution.