The aim of the study is to consider, in an unbounded domain \(\textbf{R}^n, n \ge 2\) , the problem of the existence and dynamics of a non-localized nonlinear wave equation in \(\kappa \) th-order with averaged damping using analytical methods. Using the well-known stable set method, the global solution of the system is obtained. For the asymptotic behavior of the solution, we used a variant of Nakao’s lemma in [12] with a certain modification imposed by the nature of our model. The influence of the system parameters, \(\kappa, q, p\) and initial energy on the behavior and type of existence is also discussed. We used new spaces weighted by a density function to generalize Poincaré’s inequality in an unbounded domain and obtain a Poincaré’s constant not necessarily related to the domain measurement, which is a central tool in the analysis.
