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

Anh Tuan Duong, Tuan Anh Dao

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 $$ \begin{cases} u_{t t}-Δu \geq h_1|u|^p & \text { in } V \times(0, \infty), v_{t t}-Δv \geq h_2|v|^q& \text { in } V \times(0, \infty), u=u_0;\;v=v_0 & \text { in } V \times\{0\}, u_t=u_1;\;v_t=v_1 & \text { in } V \times\{0\}, \end{cases}$$ 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.

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

Abstract

Let be a locally finite, connected and weighted graph. We study non-existence results of non-trivial, non-negative solutions of the system where , 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.
Paper Structure (4 sections, 5 theorems, 85 equations)

This paper contains 4 sections, 5 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Graph setting and Main results
  3. Proof of Theorem \ref{['th5']}
  4. Proof of Theorem \ref{['th2']}

Key Result

Theorem 2.1

Let Assumption A be satisfied. Assume that $p\geq q>1$, and $\theta_1 \geqslant 2, \theta_2 \geqslant 2$ such that $\frac{2\theta_1}{\theta_2} \geqslant 1+\alpha$. Suppose that for every $R \geqslant R_0>0$, where $E_R$ is defined in eer. Then, any non-negative weak solution of the system of e48231b must be trivial.

Theorems & Definitions (5)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Lemma 4.1