Cauchy problem for the generalized Benney-Luke equation

This paper is concerned with the study of the Cauchy problem associated with an n-dimensional generalized Benney-Luke equation, utt−mΔu−Δutt+Δ2u+α(2∇u∙∇ut+utΔu)+β∇(∣∇u∣p∇u)=0, where n=1,2,3,4. We prove the existence and the uniqueness of the global solution of the Cauchy problem for the β⩽0 case by using energy conservation law and give the existence and the nonexistence of the global solution of the Cauchy problem for the β>0 case by constructing the stable set and the unstable set.