Non-existence results for a system of wave inequalities on locally finite graphs

TL;DR

This paper establishes non-existence of non-trivial, non-negative solutions for a coupled system of wave inequalities on locally finite graphs, extending previous single-inequality results under certain volume growth conditions.

Contribution

It extends non-existence results from a single wave inequality to a coupled system on graphs, under specific geometric conditions.

Findings

No non-trivial solutions under volume growth conditions

Extension of previous results from single to coupled inequalities

Applicable to locally finite, weighted, connected graphs

Abstract

Let $V$ be a locally finite, connected and weighted graph. We study non-existence results of non-trivial, non-negative solutions of the system $$ \begin{cases} u_{t t}-\Delta u \geq h_1|v|^p & \text { in } V \times(0, \infty), v_{t t}-\Delta v \geq h_2|u|^q & \text { in } V \times(0, \infty), u=u_0;\;v=v_0 & \text { in } V \times\{0\}, u_t=u_1;\;v_t=v_1 & \text { in } V \times\{0\}, \end{cases}$$ where $p,q>1$, $h_1, h_2$ are positive potentials. Under some volume growth condition of a ball, we prove that the system has no non-trivial non-negative solutions. In particular, our result is a natural extension of that in [\textit{D.~D.~Monticelli, F.~Punzo, and J.~Somaglia. Nonexistence results for the semilinear wave equation on graphs. arXiv.2506.08697, 2025.}] from a single inequality to a system.