Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations

In this paper we consider the Cauchy problem of multidimensional generalized double dispersion equations u t t − Δ u − Δ u t t + Δ 2 u = Δ f ( u ) , where f ( u ) = a | u | p . By potential well method we prove the existence and nonexistence of global weak solution without establishing the local existence theory. And we derive some sharp conditions for global existence and lack of global existence solution.