Global existence of Cauchy problem for Boussinesq paradigm equation

In this paper we study the Cauchy problem for the Boussinesq paradigm equation utt−Δu−β1Δutt+β2Δ2u=Δf(u)utt−Δu−β1Δutt+β2Δ2u=Δf(u), where f(u)=α|u|pf(u)=α|u|p. New functionals are introduced and their sign preserving properties under the flow of the equation are studied. The existence of a global weak solution with supercritical initial energy is proved. The explicit value of the critical energy constant is found.