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Uniqueness of solution to a free boundary problem from combustion

Author(s): C. Lederman; J. L. Vázquez; N. Wolanski
Journal: Trans. Amer. Math. Soc. 353 (2001), 655-692.
MSC (1991): Primary 35K05, 35K60, 80A25
Posted: September 27, 2000
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Abstract:

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 $u(x,t)\geq 0,$ defined in a domain $\mathcal{D} \subset {\mathbb{R}}^{N}\times (0,T)$ and such that

\begin{displaymath}\Delta u+\sum a_{i}\,u_{x_{i}}-u_{t}=0\quad \text{in}\quad \mathcal{D}\cap \{u>0\}. \end{displaymath}

We also assume that the interior boundary of the positivity set, $\mathcal{D} \cap \partial \{u> \nobreak 0\}$, so-called free boundary, is a regular hypersurface on which the following conditions are satisfied:

\begin{displaymath}u=0 ,\quad -\partial u/\partial \nu = C. \end{displaymath}

Here $\nu $ 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 $\mathcal{D}$. 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: 10.1090/S0002-9947-00-02663-5
PII: S 0002-9947(00)02663-5
Keywords: Free-boundary problem, combustion, heat equation, uniqueness, classical solution, viscosity solution, limit solution
Received by editor(s): April 2, 1999
Posted: 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.
Copyright of article: Copyright 2000, American Mathematical Society

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