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A class of non-degenerate two-phase Stefan problems in several space variables. (English) Zbl 0624.35085
The singular parabolic equation $$\gamma(u)_ t-\Delta_ xu+f(u)=0$$ with either Dirichlet or nonlinear Neumann boundary conditions is considered; here u means temperature and $$\gamma(u)$$ enthalpy. Under some qualitative assumptions on the data it is shown that the mushy region can be described by two Lipschitz continuous functions $$t=\sigma^-(x)$$ and $$\sigma^+(x)$$ satisfyng $$\sigma^-\leq \sigma^+$$. This result is a consequence of a global cone condition which is proven via the maximum principle. If f satisfies the further constraint $$f(0)\geq 0$$, then $$\sigma^-=\sigma^+$$ $$(=\sigma)$$; so mushy regions cannot appear spontaneously. Moreover, temperature u satisfies the non-degeneracy property $$u(x,\sigma (x)+\epsilon)\geq C\epsilon$$.

##### MSC:
 35R35 Free boundary problems for PDEs 35K35 Initial-boundary value problems for higher-order parabolic equations 35D10 Regularity of generalized solutions of PDE (MSC2000) 35B50 Maximum principles in context of PDEs
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