A Singular Free Boundary Problem.
Abstract
The Cauchy problem is similar to the well-known one phase Stefan problem (inone space dimension). In the latter one would assume g(x) = -1 for x < 0, as well as g(x) > 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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 1983
- Accession Number
- ADA136325
Entities
People
- John A. Nohel
- K. Hollig
Organizations
- University of Wisconsin–Madison