A quasilinear parabolic singular perturbation problem

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 $u$ is a solution to the free boundary problem ${\rm div } F(\nabla u)-\partial_{t}u=0$ in $\{ u>0 \}$, $u_\nu=\alpha(\nu, M)$ on $\partial\{ u>0 \}$, in a pointwise sense and in a viscosity sense. Here $\nu$ is the inward unit spatial normal to the free boundary $\partial\{ u>0 \}$ and $M$ is a positive constant. Some of the results obtained are new even when the operator under consideration is linear.