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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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