A quasilinear parabolic singular perturbation problem

We study the following singular perturbation problem for a quasilinear uniformly parabolic operator of interest in combustion theory: div F(∇u)-∂u = β(u), where u ≥ 0, β(s) = (1/ε)β(s/ε), ε > 0, β is Lipschitz continuous, supp β = [0, 1] and β > 0 in (0, 1). We obtain uniform estimates, we pass to the limit (ε → 0) and we show that, under suitable assumptions, the limit function u is a solution to the free boundary problem div F(∇) - ∂u = 0 in {u > 0}, u = α(υ, M) on ∂{u > 0}, in a pointwise sense and in a viscosity sense. Here u denotes the derivative of u with respect to the inward unit spatial normal υ to the free boundary ∂{u > 0}, M = ∫ β(s) ds, α(υ, M) := Φ (M) and Φ(α) := - A(αυ) +αυ · F(αυ), where A(p) is such that F(p) = ∇A(p) with A(0) = 0. Some of the results obtained are new even when the operator under consideration is linear