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

Desheng Hong; Desheng Hong

doi:10.3934/math.2023922

AIMS Mathematics

2023, Volume 8, Issue 8: 18163-18173. doi: 10.3934/math.2023922

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

Desheng Hong ^,

School of Mathematics, Renmin University of China, Beijing 100872, China

Received: 23 February 2023 Revised: 14 May 2023 Accepted: 16 May 2023 Published: 26 May 2023
MSC : 35L05, 35R02, 58J45

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.
- nonlinear wave equation,
- blow up,
- locally finite graph
Citation: Desheng Hong. Blow-up of solutions for nonlinear wave equations on locally finite graphs[J]. AIMS Mathematics, 2023, 8(8): 18163-18173. doi: 10.3934/math.2023922

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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.

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