A comparison principle for the porous media equation with absorption
type: Item
instance of: scholarly article
title: A comparison principle for the porous media equation with absorption
zbMATH Open document ID: 0762.35053
DOI: 10.1016/0022-247X(92)90051-E
author: Zhengyuan Li
author: Lambertus A. Peletier
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: The authors study the following initial boundary value problem \[ u_ t=\Delta(u^ m)-\lambda u^ p,\quad u\geq 0\quad\text{in }\Omega\times\mathbb{R}^ +,\quad u(x,t)=0\quad\text{on }\partial\Omega\times\mathbb{R}^ +,\quad u(x,0)=u_ 0(x)\quad\text{in }\Omega, \] where \(m\), \(p>0\) and \(\lambda\geq 0\) are constants, \(u_ 0\) is a given bounded nonnegative function and \(\Omega\) is a bounded connected open set in \(\mathbb{R}^ n\) whose boundary is of class \(C^ 3\). A supersolution of the problem is defined as usually to be a nonnegative bounded function \(\overline u\in C(\mathbb{R}^ + :L^ 1(\Omega))\) such that \[ \int u(T)\xi(T)\geq\int u_ 0\xi(0)+\int\int(u\xi_ t+u^ m\Delta\xi- \lambda u^ p\xi),\quad\forall T>0 \] for any nonnegative test function \(\xi\) belonging to \(C^ 2(\overline\Omega\times[0,T])\) vanishing on \(\partial\Omega\times(0,+\infty)\), (Subsolutions \(\underline u\) are defined similarly, with \(\geq\) replaced by \(\leq)\). The aim of the paper is to prove a comparison principle i.e. to show that if \(\overline u(x,0)\geq\underline u(x,0)\) then \(\overline u(x,t)\geq\underline u(x,t)\). Moreover, if \(m\in(0,1)\) and \(p\geq m\), then there exists an extinction time \(T^*>0\) such that \(u=0\) in \(\overline\Omega\times(T^*,+\infty)\) and \(u>0\) in \(\Omega\times(0,T^*)\), once the initial case \(u_ 0\equiv 0\) has been excluded.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 60458
reviewed by: Mario Primicerio
MaRDI profile type: Publication
cites work: Stabilization of solutions of a degenerate nonlinear diffusion problem
cites work: Large time behaviour of solutions of the porous medium equation in bounded domains
cites work: Decay of solutions of a degenerate nonlinear diffusion equation
cites work: A positivity property of solutions of nonlinear diffusion equations
cites work: Attractivity properties of nonnegative solutions for a class of nonlinear degenerate parabolic problems
cites work: Large time behaviour of solutions of the porous media equation with absorption: the fast diffusion case
cites work: Q5734698
cites work: Continuity of solutions of a singular parabolic equation
cites work: The problem of dirichlet for quasilinear elliptic differential equations with many independent variables
full work available at URL: 0022-247x(92)90051-e
OpenAlex ID: W2005188376
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description: scientific article; zbMATH DE number 60458