Claudia Lederman

On Fast-Diffusion Equations with Infinite Equilibrium Entropy and Finite Equilibrium Mass

Abstract We extend the existing theory on large-time asymptotics for convection–diffusion equations, based on the entropy–entropy dissipation approach, to certain fast diffusion equations with uniformly convex confinement potential and finite-mass but infinite-entropy equilibrium solutions. We prove existence of a mass preserving solution of the Cauchy problem and we show exponential convergence, as , at a precise rate to the corresponding equilibrium solution in the L 1 norm. As by-product we also derive corresponding generalized Sobolev inequalities.

Uniqueness of solution to a free boundary problem from combustion

We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function u(x, t) ≥ 0, defined in a domain D ⊂ RN × (0, T ) and such that ∆u+ ∑ ai uxi − ut = 0 in D ∩ {u > 0}. We also assume that the interior boundary of the positivity set, D∩∂{u > 0}, so-called free boundary, is a regular hypersurface on which the following conditions are satisfied:

Singular Perturbation in a Nonlocal Diffusion Problem

We study a singular perturbation problem for a nonlocal evolution operator. The problem appears in the analysis of the propagation of flames in the high activation energy limit, when admitting nonlocal effects. We obtain uniform estimates and we show that, under suitable assumptions, limits are solutions to a free boundary problem in a viscosity sense and in a pointwise sense at regular free boundary points. We study the nonlocal problem both for a single equation and for a system of two equations. Some of the results obtained are new even when the operator under consideration is the heat operator.

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/e)β(s/e), e > 0, β is Lipschitz continuous, supp β = [0, 1] and β > 0 in (0, 1). We obtain uniform estimates, we pass to the limit (e → 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.