On the ν-th order solutions for nonlinear viscoelastic wave equation with averaged damping in Rⁿ

Abstract

This paper investigates the existence, uniqueness, and qualitative properties of ν-th-order solutions for a class of nonlinear viscoelastic wave equations with averaged damping in the whole space Rⁿ, n >= 3ν, ν >= 1. The model incorporates both linear memory effects and a time-averaged damping term, which captures a more realistic dissipation mechanism in complex media. By employing energy methods, stable set method, and an appropriate integral inequality, we establish the global well-posedness of higher-order solutions under suitable assumptions on the nonlinearity and initial data with minimal a priori mathematical restrictions on the parameters ν, q, p. The analysis extends previous results on lower-order formulations, providing a broader framework for understanding the dynamic response of viscoelastic materials with memory and damping.