Uniqueness of solution to a free boundary problem from combustion

Authors:
C. Lederman, J. L. Vázquez and N. Wolanski

Journal:
Trans. Amer. Math. Soc. **353** (2001), 655-692

MSC (1991):
Primary 35K05, 35K60, 80A25

DOI:
https://doi.org/10.1090/S0002-9947-00-02663-5

Published electronically:
September 27, 2000

MathSciNet review:
1804512

Full-text PDF

Abstract | References | Similar Articles | Additional Information

We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function defined in a domain and such that

We also assume that the interior boundary of the positivity set, , so-called free boundary, is a regular hypersurface on which the following conditions are satisfied:

Here denotes outward unit spatial normal to the free boundary. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of . This problem arises in combustion theory as a limit situation in the propagation of premixed flames (high activation energy limit).

The problem admits *classical* solutions only for good data and for small times. Several generalized concepts of solution have been proposed, among them the concepts of *limit* solution and *viscosity* solution. We investigate conditions under which the three concepts agree and produce a unique solution.

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Additional Information

**C. Lederman**

Affiliation:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428), Buenos Aires, Argentina

Email:
clederma@dm.uba.ar

**J. L. Vázquez**

Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Email:
juanluis.vazquez@uam.es

**N. Wolanski**

Affiliation:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428), Buenos Aires, Argentina

Email:
wolanski@dm.uba.ar

DOI:
https://doi.org/10.1090/S0002-9947-00-02663-5

Keywords:
Free-boundary problem,
combustion,
heat equation,
uniqueness,
classical solution,
viscosity solution,
limit solution

Received by editor(s):
April 2, 1999

Published electronically:
September 27, 2000

Additional Notes:
The first and third authors were partially supported by UBA grants EX071, TX47 and grant BID802/OC-AR PICT 03-00000-00137. They are members of CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas of Argentina).

The second author was partially supported by DGICYT Project PB94-0153 and HCM contract FMRX-CT98-0201.

Article copyright:
© Copyright 2000
American Mathematical Society