Global well-posedness, scattering and blow-up for the generalized Boussinesq equation in high dimensions below the ground state energy

Abstract

In this paper, we study the generalized Boussinesq equation in energy space. For spatial dimensions d≥3, nonlinear exponent α+1∈[1+4/d,1+4/(d−2)), initial data (u0,u1), we clarify the longtime behavior of the solution with energy E(u0,u1) below the ground state energy E(Qd,α,0). It depends on K(u0)=‖∇u0‖L2dα/2−2‖u0‖L22−(d−2)α/2. For K(u0)>K(Qd,α), the solution blows up in finite time. For K(u0)<K(Qd,α), the solution is global and scatters to a linear solution in energy space if u0,u1 are radial functions. To show the scattering, we use the concentration-compactness argument of Kenig-Merle [15] and the Morawetz-virial type estimate obtained in [7].