A condition for blow-up solutions to discrete semilinear wave equations on networks

Abstract

In this paper, we consider discrete semilinear wave equations $\begin{array}{ll}{u}_{tt}\left(x,t\right)={\mathrm{\Delta }}_{\omega }u\left(x,t\right)+f\left(u\left(x,t\right)\right),& \left(x,t\right)\in S\times \left(0,+\mathrm{\infty }\right),\\ u\left(x,t\right)=0,& \left(x,t\right)\in \mathrm{\partial }S\times \left[0,+\mathrm{\infty }\right),\\ u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=v_0\left(x\right),& x\in S,\end{array}$ where S is a network with boundary $\mathrm{\partial }S$. The main contribution of this work is to introduce a condition $\text{(}C\text{)}\alpha {\int }_0^{u}f\left(s\right)\mathrm{d}s\le uf\left(u\right)+\text{β}{u}^2+\gamma ,u\in \mathbb{R},$ for some $\alpha >2$ and $\text{β},\gamma \ge 0$ with $0\le \text{β}\le \left(\alpha -2\right){\lambda }_0/2$, where ${\lambda }_0$ is the first eigenvalue of discrete Laplacian ${\mathrm{\Delta }}_{\omega }$, and we use the concavity method to obtain the blow-up solutions to the above equations.

Keywords:

• Semilinear wave equations
• discrete Laplacian
• blow-up
• concavity method

2010 Mathematics Subject Classifications:

• 35L05
• 35L10

Acknowledgments

The author would like to express thanks to anonymous reviewer for their excellent suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by BK21 PLUS SNU Mathematical Sciences Division.