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arXiv · 2603.16081

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

Anh Tuan Duong·Tuan Anh Dao
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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.

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Anh Tuan Duong, Tuan Anh Dao. 2026-03-17. Non-existence results for a system of wave inequalities on locally finite graphs. https://arxiv.org/abs/2603.16081

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