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arXiv 1901.03038math.AP
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具有$q$ -反应的离散$p$ -Laplacian抛物型方程在混合边界条件下爆破解的完全刻画

A complete characterization of the blow-up solutions to discrete $p$-Laplacian parabolic equations with $q$-reaction under the mixed boundary conditions

Jaeho Hwang

备注：
29 pages, 13 figures
发表信息：
Journal of Mathematical Analysis and Applications 497 (1), 124859 (2021)
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摘要：

本文考虑了具有$q$ -反应项的离散$p$ -Laplacian抛物型方程在混合边界条件和初始条件下的解：\begin{equation*} \begin{cases} u_{t}\left(x,t\right) = Δ_{p,ω} u\left(x,t\right) +λ\left\vert u\left(x,t\right) \right\vert^{q-1} u\left(x,t\right), &\left(x,t\right) \in S \times \left(0,\infty\right), \\ μ(z)\frac{\partial u}{\partial_{p} n}(z)+σ(z)\vert u(z)\vert^{p-2}u(z)=0, &\left(x,t\right) \in \partial S \times \left[0,\infty\right), \\ u\left(x,0\right) = u_{0}(x) \geq 0, &x \in \overline{S}. \end{cases} \end{equation*}其中$p>1$，$q>0$，$λ>0$和$μ,σ$是网络$S$的边界$\partial S$上的非负函数，其中$μ(z)+σ(z)>0$，$z\in\partial S$ .这里，$Δ_{p,ω}$和$\frac{\partial ϕ}{\partial_{p} n}$分别表示离散的$p$ -拉普拉斯算子和$p$ -正规导数.参数$p>1$和$q>0$被完全刻画，以查看解何时爆破、消失或全局存在。事实上，爆破速率时，爆破确实发生。最后给出了一些数值例子来解释主要结果。

英文摘要：

In this paper, we consider discrete $p$-Laplacian parabolic equations with $q$-reaction term under the mixed boundary condition and the initial condition as follows: \begin{equation*} \begin{cases} u_{t}\left(x,t\right) = Δ_{p,ω} u\left(x,t\right) +λ\left\vert u\left(x,t\right) \right\vert^{q-1} u\left(x,t\right), &\left(x,t\right) \in S \times \left(0,\infty\right), \\ μ(z)\frac{\partial u}{\partial_{p} n}(z)+σ(z)\vert u(z)\vert^{p-2}u(z)=0, &\left(x,t\right) \in \partial S \times \left[0,\infty\right), \\ u\left(x,0\right) = u_{0}(x) \geq 0, &x \in \overline{S}. \end{cases} \end{equation*} where $p>1$, $q>0$, $λ>0$ and $μ,σ$ are nonnegative functions on the boundary $\partial S$ of a network $S$, with $μ(z)+σ(z)>0$, $z\in\partial S$. Here, $Δ_{p,ω}$ and $\frac{\partial ϕ}{\partial_{p} n}$ denote the discrete $p$-Laplace operator and the $p$-normal derivative, respectively. The parameters $p>1$ and $q>0$ are completely characterized to see when the solution blows up, vanishes, or exists globally. Indeed, the blow-up rates when blow-up does occur are derived. Also, we give some numerical illustrations which explain the main results.

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