  In this paper, we study the initial boundary value problem for the nonlinear
wave equation with combined power-type nonlinearities with variable
coefficients. The global behavior of the solutions with non-positive and
sub-critical energy is completely investigated. The threshold between the
nonexistence of global in time weak solutions and non-blowing up solutions is
found. For super-critical energy, two new sufficient conditions guaranteeing
nonexistence of global in time solutions are given. One of them is proved for
arbitrary sign of the scalar product of the initial data, while the other one
is derived only for positive sign. Uniqueness and existence of local weak
solutions are proved.
