We consider the existence, both locally and globally in time, and the blow-up of solutions for the Cauchy problem of the generalized damped multidimensional Boussinesq equation.

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We will give conditions which will guarantee the existence of global weak solutions of the Boussinesq-type equation with power-type nonlinearity γ|u| p and supercritical initial energy. By defining new functionals and using potential well method, we readdressed the initial value problem of the Boussinesq-type equation for the supercritical initial energy case.

In this Letter, the   / G G -expansion method is proposed to seek exact solutions of nonlinear evolution equations. For illustrative examples, we choose the compound KdV-Burgers equation, the compound KdV equation, the KdV-Burgers equation, the mKdV equation. The power of the employed method is confirmed.