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 \(u(x,\sigma (x)+\epsilon)\geq C\epsilon\).


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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