Global behavior of the solutions to nonlinear wave equations with combined power-type nonlinearities with variable coefficients

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