A One-Dimensional Free Boundary Problem Arising in Combustion Theory

S.N. Antontsev, A.M. Meirmanov and V.V. Yurinsky

Abstract: The free boundary problem considered in this paper arises in the mathematical theory of combustion. It consists in finding two functions $p^{\pm}(x,t)$ defined in their respective domains $\Pi^{\pm}_{T}=\bigcup_{0<t<T} \Pi^{\pm}(t)$, with $\Pi^{-}(t)=\{-1<x<R(t)\}$ and $\Pi^{+}(t)=\{R(t)<x<1\}$, that are separated by the free boundary $\Gamma_{T}=\{x=R(t)$, $t\in(0,T)\}$. In $\Pi^{\pm}_{T}$, the functions satisfy heat equations with different heat capacities, and on the free boundary they obey the conjugation conditions
$$ p^{+}(x,t)=p^{-}(x,t)=0,
\frac{\partial p^{+}(x,t)}{\partial x}-\frac{\partial p^{-}(x,t)}{\partial x}=\beta,
x=R(t).$}
Typically, the free boundary can be viewed as a model of the flame front separating the burnt and unburned domains, and $p^{\pm}$ are temperatures in these domains.
The article is dedicated to the study of the problem of existence of global-in-time classical solutions, the large-time asymptotic behavior of such solutions, and the comparison principle.
It includes some remarks on the modification of the above problem where the conjugation conditions on the free boundary specify not the jump of the temperature gradient, but the jump of its square.

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