A Condition for Blow-up solutions to Discrete $p$-Laplacian Parabolic Equations under the mixed boundary conditions on Networks

Abstract

The purpose of this paper is to investigate a condition \begin{equation*} (C_{p}) \hspace{1cm} α\int_{0}^{u}f(s)ds \leq uf(u)+βu^{p}+γ,\,\,u>0 \end{equation*} for some $\alpha >2$, $\gamma >0$, and $0\le \beta \le \frac{\left(\alpha -p\right){\lambda }_{p,0}}{p}$, where $p>1$ and ${\lambda }_{p,0}$ is the first eigenvalue of the discrete $p$-Laplacian ${\Delta }_{p,\omega }$. Using the above condition, we obtain blow-up solutions to discrete $p$-Laplacian parabolic equations \begin{equation*} \begin{cases} u_{t}\left(x,t\right)=Δ_{p,ω}u\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in S\times\left(0,+\infty\right), μ(z)\frac{\partial u}{\partial_{p} n}(x,t)+σ(z)|u(x,t)|^{p-2}u(x,t)=0, & \left(x,t\right)\in\partial S\times\left[0,+\infty\right), u\left(x,0\right)=u_{0}\geq0(nontrivial), & x\in S, \end{cases} \end{equation*} on a discrete network $S$, where $\frac{\partial u}{{\partial }_{p}n}$ denotes the discrete $p$-normal derivative. Here, $\mu$ and $\sigma$ are nonnegative functions on the boundary $\partial S$ of $S$, with $\mu (z)+\sigma (z)>0$, $z\in \partial S$. In fact, it will be seen that the condition $(C_p)$, the generalized version of the condition $(C)$, improves the conditions known so far.

22 pages. arXiv admin note: substantial text overlap with arXiv:1706.03494