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 $\mathrm{div}F(\nabla u)-{\partial }_{t}u=0$ 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.