The Cauchy problem for a class of the multidimensional Boussinesq-type equation

In this paper, we study the Cauchy problem for a class of the multidimensional Boussinesq-type equation utt−Δu+Δ2u+Δ2utt=Δf(u), where f(u)=±a|u|pf(u)=±a|u|p or −a|u|p−1u−a|u|p−1u, a>0a>0 is a constant. First, we establish a local existence theorem for the solution. Then, for m=1(n=1), m=2,3,4(n≤3), we prove the existence of a global HmHm solution. Finally, we prove the global nonexistence and finite-time blow up of the solution.