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Uniqueness of limit solutions to a free boundary problem from combustion. (English) Zbl 1083.35039
Let $$f_\varepsilon(s)=\frac 1 {\varepsilon^2}f\left(\frac s \varepsilon \right)$$, where $$f$$ is a Lipschitz continuous function with $$f(s)>0$$ if $$z<1$$ and $$f(s)=0$$ if $$z\geq 1$$. The authors consider the following problem arising in combustion theory: $\Delta u^\varepsilon -u^\varepsilon_t=Y^\varepsilon f_\varepsilon(u^\varepsilon) \;\text{ in } D,$ $\Delta Y^\varepsilon -Y^\varepsilon_t=Y^\varepsilon f_\varepsilon(u^\varepsilon) \;\text{ in } D,$ where $$D\subset\mathbb R^{n+1}$$. They investigate the convergence of $$u^\varepsilon$$ to the solution of a free boundary problem $\Delta u-u_t=0, \;\;in \;\;\{ u>0 \},$ $| \nabla u| =\sqrt{2M(x,t)}\;\;on\;\;\partial \{ u>0 \},$ with a prescribed function $$M(x,t)$$.

##### MSC:
 35K05 Heat equation 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 80A25 Combustion
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