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

Abstract

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 $ \partial^2_t u = \Delta u + f(u) $ 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.

Keywords

• nonlinear wave equation
• blow up
• locally finite graph