Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A free-boundary problem for the heat equation arising in flame propagation

Authors: Luis A. Caffarelli and Juan L. Vázquez
Journal: Trans. Amer. Math. Soc. 347 (1995), 411-441
MSC: Primary 35K57; Secondary 35R35, 80A22, 80A25
MathSciNet review: 1260199
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Abstract: We introduce a new free-boundary problem for the heat equation, of interest in combustion theory. It is obtained in the description of laminar flames as an asymptotic limit for high activation energy. The problem asks for the determination of a domain in space-time, $ \Omega \subset {{\mathbf{R}}^n} \times (0,T)$, and a function $ u(x,t) \geqslant 0$ defined in $ \Omega $, such that $ {u_t} = \Delta u$ in $ \Omega ,\;u$ takes certain initial conditions, $ u(x,0) = {u_0}(x)$ for $ x \in {\Omega _0} = \partial \Omega \cap \{ t = 0\} $, and two conditions are satisfied on the free boundary $ \Gamma = \partial \Omega \cap \{ t > 0\} :u = 0$ and $ {u_\nu } = - 1$, where $ {u_\nu }$ denotes the derivative of $ u$ along the spatial exterior normal to $ \Gamma $. We approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data.

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Keywords: Heat equation, free-boundary problem, combustion, regularization method, weak solutions
Article copyright: © Copyright 1995 American Mathematical Society