Non-existence results for a system of wave inequalities on locally finite graphs
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908902590775296 |
|---|---|
| author | Duong, Anh Tuan Dao, Tuan Anh |
| author_facet | Duong, Anh Tuan Dao, Tuan Anh |
| contents | Let $V$ be a locally finite, connected and weighted graph. We study non-existence results of non-trivial, non-negative solutions of the system $$ \begin{cases}
u_{t t}-Δu \geq h_1|v|^p & \text { in } V \times(0, \infty),
v_{t t}-Δv \geq h_2|u|^q & \text { in } V \times(0, \infty),
u=u_0;\;v=v_0 & \text { in } V \times\{0\},
u_t=u_1;\;v_t=v_1 & \text { in } V \times\{0\},
\end{cases}$$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16081 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-existence results for a system of wave inequalities on locally finite graphs Duong, Anh Tuan Dao, Tuan Anh Analysis of PDEs Primary: 35A01, 35A02, 35R45. Secondary: 35R02 Let $V$ be a locally finite, connected and weighted graph. We study non-existence results of non-trivial, non-negative solutions of the system $$ \begin{cases} u_{t t}-Δu \geq h_1|v|^p & \text { in } V \times(0, \infty), v_{t t}-Δv \geq h_2|u|^q & \text { in } V \times(0, \infty), u=u_0;\;v=v_0 & \text { in } V \times\{0\}, u_t=u_1;\;v_t=v_1 & \text { in } V \times\{0\}, \end{cases}$$ 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. |
| title | Non-existence results for a system of wave inequalities on locally finite graphs |
| topic | Analysis of PDEs Primary: 35A01, 35A02, 35R45. Secondary: 35R02 |
| url | https://arxiv.org/abs/2603.16081 |