On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium
type: Item
instance of: scholarly article
title: On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium
zbMATH Open document ID: 0634.35042
DOI: 10.1016/0022-0396(87)90125-2
author: Robert Kersner
author: J. I. Díaz
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: The paper under review addresses the equation \[ u(x,t)_t=\Phi(u(x,t))_{xx} +b(u(x,t))_x \] where \(\Phi\) and \(b\) are real continuous functions, \(t>0\), and \(x\) varies (i) on the whole real line, (ii) on a (bounded) interval, (iii) on a half-line; in the last two cases, non- homogeneous Dirichlet boundary conditions are provided at the finite endpoint(s); in any case, the problem is supplemented with initial data. This equation occurs in modelling infiltration or evaporation in a porous medium: for this reason, \(\Phi(u)\) is assumed to ``degenerate'', i.e., to have a vanishing derivative at \(u=0\). The author first provide sufficient conditions for the existence of limit solutions, namely objects obtained as pointwise limits of approximate classical solutions, namely solutions of nondegenerate problems in appropriate cut-off domains. Then they give further conditions in order such limit solution to be weak solutions. In this context, disproving a conjecture put forward in some previous literature on the subject, they establish the dependency of the modulus of continuity of the weak solution of the transport term \(b(u)_x\). Finally, the author prove uniqueness results, providing at the same time a thorough and unified discussion of the result in this direction obtained so far.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35A35
Mathematics Subject Classification ID: 35D99
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 4032215
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: initial data
zbMATH Keywords: infiltration
zbMATH Keywords: evaporation
zbMATH Keywords: porous medium
zbMATH Keywords: existence of limit solutions
zbMATH Keywords: weak solutions
zbMATH Keywords: modulus of continuity
zbMATH Keywords: uniqueness
MaRDI profile type: Publication
full work available at URL: 0022-0396(87)90125-2
OpenAlex ID: W1987310399
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description: scientific article; zbMATH DE number 4032215