Let $V$ be a locally finite, connected and weighted graph. We study non-existence results of non-trivial, non-negative solutions of the system $$ \begin{ cases} u_{ t t}-\Delta u \geq h_1|u|^p & \text { in } V \times(0, \infty), v_{ t t}-\Delta v \geq h_2|v|^q& \text { in } V \times(0, \infty), u=u_0;\;v=v_0 & \text { in } V \times\{ 0\}, u_t=u_1;\;v_t=v_1 & \text { in } V \times\{ 0\}, \end{ cases}$$ 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.
Categories: math.AP