Finite time blow - up of solutions for system of κth order wave equations with nonlinear averaged damping and sources

We consider basic relationships between the initial energy and the exponent of nonlinear sources in a nonlinear coupled system of κth order wave equations with nonlinear averaged damping, and we demonstrate the global nonexistence of solutions. This approach is a variant of the method of nonlinear functional analysis with a contradiction argument, which is one of the tools for proving the blow - up of solutions for nonlinear partial differential equations. A new class of nonlinear coupled wave equations with nonlinear sources is given in high-order functional spaces.