A Singular Free Boundary Problem.
Technical summary rept.,
WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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The Cauchy problem is similar to the well-known one phase Stefan problem inone space dimension. In the latter one would assume gx -1 for x 0, as well as gx 0 for x 0, so that g would have a jump discontinuity at x 0. Our assumptions on the initial data g yield a different behavior of the solution v and of the resulting free boundary. Indeed, the free boundary is not infinitely differentiable at t 0, contrary to the situation for the classical Stefan problem. This problem also serves as a prototype of nonlinear parabolic problems which arise as monotone convexifications of nonlinear diffusion equations with nonmonotone constitutive functions phi. That analysis shows the existence of infinitely many solutions v of the nonmonotone problem each having v bounded, but oscillating more and more rapidly as t infinity 0. Thus each solution v exhibits phase changes. Numerical experiments further suggest the conjecture that the physically correct solution of the nonmonotone problem is the one which for t 0 sufficiently large approaches the unique solution of the appropriately related convexified monotone problem. This paper is another step towards the understanding of this intriguing phenomenon.
- Theoretical Mathematics