Well-Posedness of Solutions for the Sixth-Order Boussinesq Equation with Linear Strong Damping and Nonlinear Source

The object of this paper is to study a sixth-order Boussinesq equation with dispersive, linear strong damping and nonlinear source by using potential well methods, including the following aspects: firstly, the local well-posedness of the solutions is studied; secondly, the global existence and the finite time blow-up conditions are studied at two different initial energy levels by using the relationship between the initial energy and the depth of the potential well; thirdly, a blow-up condition independent of the depth of the potential well is established and by using of this condition, the existence of blow-up solutions at arbitrary initial energy level is studied; finally, the upper bound estimation of blow-up time and some necessary and sufficient conditions for existing finite time blow-up solutions are established.