A nonlinear oblique derivative boundary value problem for the heat equation: analogy with the porous medium equation
type: Item
instance of: scholarly article
title: A nonlinear oblique derivative boundary value problem for the heat equation: analogy with the porous medium equation
zbMATH Open document ID: 1016.35038
DOI: 10.1016/S0294-1449(01)00087-7
author: J. M. Roquejoffre
author: Luis Caffarelli
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2003-08-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78539
review text: This paper studies the heat equation in the upper half-plane with diffuison in only one direction together with an oblique derivative boundary condition. Global existence for the Cauchy problem is proved and qualitative properties of the solutions are stated: this means compact support for solutions, finite speed of propagation, free boundary relations and convergence to self-similar solutions as asymptotic behavior. This problem exhibits a strong analogy with the porous media equation. The problem arises in modelling a plasma opening switch. The main result gives existence of a global classical solution for suitable initial data together with a \(C^{1,\alpha}\) free boundary and the smoothness of the ``pressure'' \(u^2\) for the solution \(u\). Existence is proved by an approximation argument which is developed in Section 2, where interesting comparison results are also obtained. Finally, existence of a unique viscosity solution is obtained; moreover, there is a comparison result for viscosity solutions. Abel type integral operators are used here. Section 3 deals with travelling waves and self-similar solutions, and it is proved that the latter are also viscosity solutions. The proof that the interface is Lipschitz and nondegenerate is given in Section 4, and also that similar properties hold for the pressure as well. Some Harnack inequalities are also proved. The \(C^{1,\alpha}\) property of the free boundary is proved in Section 5 by using an iteration technique of Caffarelli-Wolanski and, finally, the asymptotic behavior and the free boundary relations are studied in Section 6.
reviewed by: Jesús Hernández
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1785832
zbMATH Keywords: travelling waves
zbMATH Keywords: viscosity solutions
zbMATH Keywords: Harnack inequalities
zbMATH Keywords: compact support for solutions
zbMATH Keywords: finite speed of propagation
zbMATH Keywords: free boundary relations
zbMATH Keywords: convergence to self-similar solutions
MaRDI profile type: Publication
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OpenAlex ID: W2008579951
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description: scientific article; zbMATH DE number 1785832