Existence and Dynamics of $$\kappa $$th-order Solution for Nonlinear Wave Equation with Averaged Damping in Weighted Spaces

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.