A nonexistence result for a nonlinear wave equation with damping on a Riemannian manifold

In this paper, we study the global nonexistence of solutions to a nonlinear wave equation with critical potential V(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V(x)$\end{document} on a Riemannian manifold, the form of which is more general than those in (Todorova and Yordanov in C. R. Acad. Sci., Sér. 1 Math. 300:557-562, 2000). The way we follow is motivated by the work of Qi S. Zhang (C. R. Acad. Sci., Sér. 1 Math. 333:109-114, 2001). We also prove the local existence and uniqueness result.