A parabolic free boundary problem with double pinning Original Research Article

We study the Cauchy problem for the equation ∂tuε−Δuε=−βε(uε) in (0,∞)×Rn as ε→0, where the nonlinearity βε is assumed to converge to a measure concentrated at 0. In this paper we allow for sign changes of βε and uε. The solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. We show that each limit of uε is a solution of the free boundary problem ∂tu−Δu=0 in {u>0}∩(0,∞)×Rn,|∇u+|2−|∇u−|2=g on (∂{u>0}∪∂{u<0})∩((0,∞)×Rn) in the sense of domain variations. Depending on the structure of the nonlinearity View the MathML source the function g in the condition on the free boundary need not be a constant.