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**On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium.**
*(English)*
Zbl 0634.35042

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.

\[ u(x,t)_t=\Phi(u(x,t))_{xx} +b(u(x,t))_x \] 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.

Reviewer: P. de Mottoni (Roma)

### MSC:

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

35A35 | Theoretical approximation in context of PDEs |

35D99 | Generalized solutions to partial differential equations |

35D30 | Weak solutions to PDEs |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35K65 | Degenerate parabolic equations |

### Keywords:

Dirichlet boundary conditions; initial data; infiltration; evaporation; porous medium; existence of limit solutions; weak solutions; modulus of continuity; uniqueness
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\textit{J. I. Diaz} and \textit{R. Kersner}, J. Differ. Equations 69, 368--403 (1987; Zbl 0634.35042)

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