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

In this paper, we study the generalized Boussinesq equation in energy space. For spatial dimensions d > 3, nonlinear exponent alpha + 1 E [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,alpha, 0). It depends on K(u0) = IIou0IId alpha/2-2 L2 IIu0II2-(d-2)alpha/2. For K(u0) > K(Qd,alpha), the solution blows up in finite time. For K(u0) < K(Qd,alpha), 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].