Existence and approximation of weak solutions of the Stefan problem with nonmonotone nonlinearities

Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 506)


Consider the equation, in the distribution sense, for the temperature in a two-phase multidimensional Stefan problem
$$\frac{{\partial u}}{{\partial t}} - \nabla .(k(u) \nabla u) + g(u) = f$$
on a space-time domain D=(O,T) × Ω with specified initial and boundary conditions and enthalpy discontinuity across the free boundary. Here the conductivity coefficient k is a positive function with compact range, defined and continuous on R except at O, and g is a Lipschitz body heating function, frequently encountered in welding problems, which is not assumed monotone. (We may take g such that g(u)u≥0).

Implicit two level time discretizations are employed in transformed versions of (1), giving a (finite) sequence of nonlinear elliptic boundary value problems (for each δt) which are solved by a Galerkin method. A subsequence of the step functions constructed on D is shown to converge weakly to a weak solution of the transformed equation. If, in addition, g is monotone, the entire sequence is strongly convergent to the unique solution.


Weak Solution Stefan Problem Unique Weak Solution Quasilinear Parabolic Equation Move Boundary Problem 
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