摘要：In this paper, we investigate the condition$$(C_{p})\quad \alpha \int _{0}^{u}f(s)\,ds \leq uf(u)+\beta u^{p}+\gamma ,\quad u>0 $$ for some $\alpha >2$, $\gamma >0$, and $0\leq \beta \leq \frac{ (\alpha -p ) \lambda _{p,0}}{p}$, where $p>1$, and $\lambda _{p,0}$ is the first eigenvalue of the discrete p-Laplacian $\Delta _{p,\omega }$. Using this condition, we obtain blow-up solutions to discrete p-Laplacian parabolic equations$$ \textstyle\begin{cases} u_{t} (x,t )=\Delta _{p,\omega }u (x,t )+f(u(x,t)), & (x,t )\in S\times (0,+\infty ), \\ \mu (z)\frac{\partial u}{\partial _{p} n}(x,t)+\sigma (z) \vert u(x,t) \vert ^{p-2}u(x,t)=0, & (x,t )\in \partial S\times [0,+\infty ), \\ u (x,0 )=u_{0}\geq 0\quad (\mbox{nontrivial}), & x\in S, \end{cases} $$ on a discrete network S, where $\frac{\partial u}{\partial _{p}n}$ denotes the discrete p-normal derivative. Here μ and σ are nonnegative functions on the boundary ∂S of S with $\mu (z)+\sigma (z)>0$, $z\in \partial S$. In fact, we will see that condition $(C_{p})$ improves the conditions known so far..
关键词：Discrete p -Laplacian ; Semilinear parabolic equation ; Blow-up ;
