A Wave Inequality with Convolution Nonlinearities

We consider a wave inequality in an exterior domain of RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{N}$$\end{document}, involving the product of two nonlinear convolution terms. The problem is considered under an inhomogeneous Dirichlet-type boundary condition. We establish sufficient conditions depending on the parameters of the problem, under which we have existence/nonexistence of weak solutions.