JournalsifbVol. 10 , No. 4DOI 10.4171/ifb/197

A quasilinear parabolic singular perturbation problem

  • Claudia Lederman

    Universidad de Buenos Aires, Argentina
  • Dietmar Oelz

    Universität Wien, Austria
A quasilinear parabolic singular perturbation problem cover

Abstract

We study a singular perturbation problem for a quasilinear uniformly parabolic operator of interest in combustion theory. We obtain uniform estimates, we pass to the limit and we show that, under suitable assumptions, the limit function uu is a solution to the free boundary problem divF(u)tu=0{\rm div } F(\nabla u)-\partial_{t}u=0 in {u>0}\{ u>0 \}, uν=α(ν,M)u_\nu=\alpha(\nu, M) on {u>0}\partial\{ u>0 \}, in a pointwise sense and in a viscosity sense. Here ν\nu is the inward unit spatial normal to the free boundary {u>0}\partial\{ u>0 \} and MM is a positive constant. Some of the results obtained are new even when the operator under consideration is linear.