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} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{p}+\gamma,\,\,u>0 \end{equation*} for some $\alpha>2$, $\gamma>0$, and $0\leq\beta\leq\frac{\left(\alpha-p\right)\lambda_{p,0}}{p}$, 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)=\Delta_{p,\omega}u\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in S\times\left(0,+\infty\right), \mu(z)\frac{\partial u}{\partial_{p} n}(x,t)+\sigma(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.