The porous medium equation with nonlinear absorption and moving boundaries
type: Item
instance of: scholarly article
title: The porous medium equation with nonlinear absorption and moving boundaries
zbMATH Open document ID: 0611.35047
DOI: 10.1007/BF02764879
author: Semion Gutman
author: Robert H. jun. Martin
published in: Israel Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Dirichlet and Neumann boundary value problem for the nonlinear parabolic equation \[ (1)\quad u_ t=[\phi (u)]_{xx}+f(u) \] are analysed, when \(\phi\) : \(R\to R\) is \(C^ 2\) and \(\phi (0)=0\) and \(\phi '(r)>0\) if \(r\neq 0\). The main examples are \(\phi (r)=dr^ m\), \(d>0\), \(m\geq 1\). The paper is divided into three sections. The first part deals with the existence of (generalized) solutions to the Dirichlet and Neumann problems with moving boundaries by using the theory of nonlinear semigroups. The solutions are constructed using approximations and the Crandall-Ligett theorem for the generation of semigroups in general Banach spaces. The second part deals with the asymptotic behavior of solutions, and the third part gives some differentiability properties of the generalized solutions. It is shown that the general types of behavior for the solution of equation (1) is similar to the semilinear case.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 76S05
zbMATH DE Number: 3987820
zbMATH Keywords: porous medium
zbMATH Keywords: nonlinear absorption
zbMATH Keywords: Dirichlet
zbMATH Keywords: Neumann
zbMATH Keywords: existence
zbMATH Keywords: moving boundaries
zbMATH Keywords: nonlinear semigroups
zbMATH Keywords: approximations
zbMATH Keywords: Crandall-Ligett theorem
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: differentiability properties
zbMATH Keywords: generalized solutions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02764879
OpenAlex ID: W2034141830
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description: scientific article; zbMATH DE number 3987820