## Abstract and Applied Analysis

### Blow-Up Solutions and Global Solutions to Discrete $p$-Laplacian Parabolic Equations

#### Abstract

We discuss the conditions under which blow-up occurs for the solutions of discrete $p$-Laplacian parabolic equations on networks $S$ with boundary $\partial S$ as follows: ${u}_{t}(x,t)={\mathrm{\Delta }}_{p,\omega }u(x,t)+\lambda |u(x,t){|}^{q-1}u(x,t)$, $(x,t)\in S{\times}(0,+\mathrm{\infty })$; $u(x,t)=0$, $(x,t)\in \partial S{\times}(0,+\mathrm{\infty })$; $u(x,0)={u}_{0}\ge 0,$ $x\in \overline{S}$, where $p>1$, $q>0$, $\lambda >0$, and the initial data ${u}_{0}$ is nontrivial on $S$. The main theorem states that the solution $u$ to the above equation satisfies the following: (i) if $0 and $q>1$, then the solution blows up in a finite time, provided ${\overline{u}}_{0}>{({\omega }_{0}/\lambda )}^{1/(q-p+1)}$, where ${\omega }_{0}:={\mathrm{max}}_{x\in S}{\sum }_{y\in \overline{S}}\mathrm{‍}\omega (x,y)$ and ${\overline{u}}_{0}:={\text{max}}_{x\in S}{\mathrm{ u}}_{0}(x)$; (ii) if $0, then the nonnegative solution is global; (iii) if $1, then the solution is global. In order to prove the main theorem, we first derive the comparison principles for the solution of the equation above, which play an important role throughout this paper. Moreover, when the solution blows up, we give an estimate for the blow-up time and also provide the blow-up rate. Finally, we give some numerical illustrations which exploit the main results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 351675, 11 pages.

Dates
First available in Project Euclid: 27 February 2015

https://projecteuclid.org/euclid.aaa/1425049883

Digital Object Identifier
doi:10.1155/2014/351675

Mathematical Reviews number (MathSciNet)
MR3285158

Zentralblatt MATH identifier
07022208

#### Citation

Chung, Soon-Yeong; Choi, Min-Jun. Blow-Up Solutions and Global Solutions to Discrete $p$ -Laplacian Parabolic Equations. Abstr. Appl. Anal. 2014 (2014), Article ID 351675, 11 pages. doi:10.1155/2014/351675. https://projecteuclid.org/euclid.aaa/1425049883