Hyperbolic Inequalities Involving Nonlinear Convolution Terms on Riemannian Manifolds

We consider a class of hyperbolic inequalities with nonlinear convolution terms on complete noncompact Riemannian manifolds. We establish sufficient conditions depending on the geometry of the manifold and the parameters of the problem, for which there is no nontrivial global weak solution. Moreover, in the Euclidean case, the existence of positive solutions is investigated. Our obtained results are new even in the Euclidean case.