Non-existence results for a system of wave inequalities on locally finite 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 $$\{\begin{array}{llccc}{u}_{tt}-\Delta u\ge h_1|v{|}^p& \text{ in }V\times (0,\infty ),v_{tt}-\Delta v\ge h_2|u{|}^q& \text{ in }V\times (0,\infty ),u=u_0;v=v_0& \text{ in }V\times \{0\},u_t=u_1;v_t=v_1& \text{ in }V\times \{0\},\end{array}$$ 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.

17 pages