Blow up results and lifespan estimates for nonlinear damped wave equations on weighted graphs

In this article, we are interested in studying the Cauchy problems for nonlinear damped wave equations and their systems on a weighted graph. Our main purpose is two-fold, namely, under certain conditions for volume growth of a ball and the initial data we would like to not only prove nonexistence of global (in time) weak solutions but also indicate lifespan estimates for local (in time) weak solutions when a blow-up phenomenon in finite time occurs. Throughout the present paper, we will partially give a positive answer for the optimality of our results by an application to the $n$-dimensional integer lattice graph $\Z^n$ to recover the well-known results in the Euclidean setting.