Blow-up solutions and global solutions to discrete \(p\)-Laplacian parabolic equations
type: Item
instance of: scholarly article
title: Blow-up solutions and global solutions to discrete \(p\)-Laplacian parabolic equations
zbMATH Open document ID: 1469.35133
DOI: 10.1155/2014/351675
author: Soon-Yeong Chung
author: Min-Jun Choi
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: 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: \(u_t(x,t)=\Delta_{p,\omega}u(x,t)+\lambda|u(x,t)|^{q-1}u(x,t)\), \((x,t)\in S\times(0,+\infty)\); \(u(x,t)=0\), \((x,t)\in\partial S\times(0,+\infty)\); \(u(x,0)=u_0\geq0\), \(x\in\overline S\), 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 \(0<p-1<q\) and \(q>1\), then the solution blows up in a finite time, provided \(\overline u_0>(\omega_0/\lambda)^{1/(q-p+1)}\), where \(\omega_0:=\max_{x\in S}\sum_{y\in\overline S}\omega(x,y)\) and \(\overline u_0:=\max_{x\in S}u_0(x)\); (ii) if \(0<q\leq1\), then the nonnegative solution is global; (iii) if \(1<q<p-1\), 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.
Mathematics Subject Classification ID: 35K92
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 7022208
Wikidata QID: Q59036439
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/351675
OpenAlex ID: W1986071571
cites work: Asymptotic behavior of blowup solutions of a parabolic equation with the \(p\)-Laplacian
cites work: A note on blow‐up for fast diffusive p‐Laplacian with sources
cites work: Q4804250
cites work: Critical extinction and blow‐up exponents for fast diffusive p‐Laplacian with sources
cites work: Q3610076
cites work: Extinction and positivity of the solutions of the heat equations with absorption on networks
cites work: Extinction and positivity of solutions of the \(p\)-Laplacian evolution equation on networks
cites work: Q5691133
cites work: $\omega$-Harmonic Functions and Inverse Conductivity Problems on Networks
cites work: Q3907599
cites work: Positive solutions for discrete boundary value problems involving the p-Laplacian with potential terms
cites work: The Dirichlet boundary value problems forp-Schrödinger operators on finite networks
description: scientific article; zbMATH DE number 7022208