Blow-up of solutions for nonlinear wave equations on locally finite graphs

Abstract

<abstract><p>Let $ G = (V, E) $ be a local finite connected weighted graph, $ \Omega $ be a finite subset of $ V $ satisfying $ \Omega^\circ\neq\emptyset $. In this paper, we study the nonexistence of the nonlinear wave equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial^2_t u = \Delta u + f(u) $\end{document} </tex-math></disp-formula></p> <p>on $ G $. Under the appropriate conditions of initial values and nonlinear term, we prove that the solution for nonlinear wave equation blows up in a finite time. Furthermore, a numerical simulation is given to verify our results.</p></abstract>