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arXiv 1901.03075math.AP
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离散$p$ -Laplacian抛物型方程在网络上混合边界条件下解爆破的条件

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

Soon-Yeong Chung, Min-Jun Choi, Jaeho Hwang

备注：
22 pages. arXiv admin note: substantial text overlap with arXiv:1706.03494
发表信息：
Boundary Value Problems 2019 (1), 180
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摘要：

本文的目的是研究一个条件\begin{equation*} (C_{p}) \hspace{1cm} α\int_{0}^{u}f(s)ds \leq uf(u)+βu^{p}+γ,\,\,u>0 \end{equation*}为某些$α>2$，$γ>0$，和$0\leqβ\leq\frac{\left(α-p\right)λ_{p,0}}{p}$，其中$p>1$和$λ_{p,0}$是离散的$p$-Laplacian $Δ_{p,ω}$的第一特征值.利用上述条件，我们得到了离散网络$S$上离散$p$ -Laplacian抛物型方程\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*}的爆破解，其中$\frac{\partial u}{\partial_{p}n}$表示离散$p$ -正规导数.这里，$μ$和$σ$是$S$的边界$\partial S$上的非负函数，其中$μ(z)+σ(z)>0$，$z\in \partial S$。事实上，可以看出，条件$(C_{p})$，条件$(C)$的广义版本，改进了迄今为止已知的条件。

英文摘要：

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 $α>2$, $γ>0$, and $0\leqβ\leq\frac{\left(α-p\right)λ_{p,0}}{p}$, where $p>1$ and $λ_{p,0}$ is the first eigenvalue of the discrete $p$-Laplacian $Δ_{p,ω}$. 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, $μ$ and $σ$ are nonnegative functions on the boundary $\partial S$ of $S$, with $μ(z)+σ(z)>0$, $z\in \partial S$. In fact, it will be seen that the condition $(C_{p})$, the generalized version of the condition $(C)$, improves the conditions known so far.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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