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(∇uε)-∂tuε = βε(uε), where uε ≥ 0, βε(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 es...
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2008
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_14639963_v10_n4_p447_Lederman http://hdl.handle.net/20.500.12110/paper_14639963_v10_n4_p447_Lederman |
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| Sumario: | We study the following singular perturbation problem for a quasilinear uniformly parabolic operator of interest in combustion theory: div F(∇uε)-∂tuε = βε(uε), where uε ≥ 0, βε(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) - ∂tu = 0 in {u > 0}, uυ = α(υ, M) on ∂{u > 0}, in a pointwise sense and in a viscosity sense. Here uυ 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) := Φv -1 (M) and Φv(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. © European Mathematical Society 2008. |
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