Singular Perturbation in a Nonlocal Diffusion Problem

Abstract

Abstract. 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 local monotonicity formula for an inhomogenous singular perturbation problem and applications

“…All these cases can happen as was shown with examples in [13]. So that our results are really sharp.…”

“…For instance, for V ε = 0 they were proved in [13] in the one phase case (i.e. u ε ≥ 0) and in [14] in the stationary two phase case.…”

“…Observe that in [13] it was proved in the one phase case that solutions u ε to P ε (f ε ) with uniformly bounded Lip (1, 1/2) norm and…”

Asymptotic behavior for a nonlocal diffusion equation with absorption and nonintegrable initial data. The supercritical case

“…The uniqueness of bounded solutions follows from the comparison principle for bounded solutions. (See [16], Proposition 2.2 with θ = 0 for the case of smooth solutions. In order to get the result in our case we just have to approximate the initial datum by smooth functions that are uniformly bounded.…”

“…Since positive constants are supersolutions and negative constants are subsolutions to problem (3.1), and the function τ → |τ | p−1 τ is locally Lipschitz, the comparison principle (see, for instance, [16]) implies that the fixed point u is bounded by u 0 ∞ . Therefore, the solution can be extended for all times.…”