Universal self-similarity of porous media equation with absorption: the critical exponent case.
type: Item
instance of: scholarly article
title: Universal self-similarity of porous media equation with absorption: the critical exponent case.
zbMATH Open document ID: 1058.35130
published in: Journal of Differential Equations
publication date: 2004-03-29T00:00:00Z
review text: The authors consider the \(N\)-dimensional Cauchy problem \[ \begin{cases} u_t= \Delta_x u^m- u^q\quad &\text{in }\mathbb{R}^N\times (0,\infty),\\ u(x,0)= u_0(x)\geq 0\quad & \text{in }\mathbb{R}^N,\;u_0\in L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N),\end{cases} \] where \(m> 1\) and \(q= q_{\text{crit}}\equiv m+{2\over N}\). They study the global dynamics of general solution with \(L^1\)-initial data having mild decay as \(|x|\to\infty\). They derive the exact spatio-temporal profile of a solution when \(t\to\infty\).
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 2062887
zbMATH Keywords: large time behavior
zbMATH Keywords: exact spatio-temporal profile
zbMATH Keywords: self-similar solutions
author: Yuanwei Qi
author: Xu-Dong Liu
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: j.jde.2003.10.022
OpenAlex ID: W2021689540
cites work: A very singular solution of the heat equation with absorption
cites work: Q3940162
cites work: Universality in blow-up for nonlinear heat equations
cites work: Renormalization group and asymptotics of solutions of nonlinear parabolic equations
cites work: Global large time self-similarity of a thermal-diffusive combustion system with critical nonlinearity
cites work: A boundary modulus of continuity for a class of singular parabolic equations
cites work: Q3818689
cites work: Large time behavior of solutions of a dissipative semilnear heat equation
cites work: The Asymptotic Behavior of Gas in an n-Dimensional Porous Medium
cites work: ON ASYMPTOTIC “EIGENFUNCTIONS” OF THE CAUCHY PROBLEM FOR A NONLINEAR PARABOLIC EQUATION
cites work: Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach
cites work: Large time behavior of the solutions of a semilinear parabolic equation in \(\mathbb R^ n\)
cites work: Asymptotic behaviour of solutions of some semilinear parabolic problems
cites work: Large time behaviour of solutions of the porous media equation with absorption
cites work: Q4051019
cites work: О Некоторых свойствах обобщенных решений квазилинейных вырождающихся параболических уравнений
cites work: Large time behaviour of solutions of the porous media equation with absorption: the fast diffusion case
DOI: 10.1016/J.JDE.2003.10.022
Recommended article: On very singular self-similar solutions for the porous media equation with absorption
Recommended article: Q4919109
Recommended article: Q4897913
Recommended article: Q3490343
Recommended article: Q3193956
Recommended article: Q2956371
Recommended article: On the critical exponents for porous medium equation with a localized reaction in high dimensions
Recommended article: Stochastic porous media equations and self-organized criticality: Convergence to the critical state in all dimensions
Recommended article: Stochastic porous media equations and self-organized criticality
Recommended article: Q3461000
description: scientific article; zbMATH DE number 2062887