quenching time for a nonlocal diffusion problem with large initial data

Quenching Time for a Nonlocal Diffusion Problem with Large Initial Data

Large Time Behavior for a Nonlocal Diffusion Equation with Absorption and Bounded Initial Data

Abstract. We study the large time behavior of nonnegative solutions of the Cauchy problem ut = ∫ J(x − y)(u(y, t) − u(x, t)) dy − u, u(x, 0) = u0(x) ∈ L , where |x|u0(x) → A > 0 as |x| → ∞. One of our main goals is the study of the critical case p = 1 + 2/α for 0 < α < N , left open in previous articles, for which we prove that t|u(x, t) − U(x, t)| → 0 where U is the solution of the heat equati...

Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data: The subcritical case

This is a joint work with A. Salort and N. Wolanski. We continue our study of the large time behavior of the bounded solution to the nonlocal diffusion equation with absorption { ut = Lu− u in R × (0,∞), u(x, 0) = u0(x) in R , where p > 1, u0 ≥ 0 and bounded and Lu(x, t) = ∫ J(x− y) (u(y, t)− u(x, t)) dy with J ∈ C∞ 0 (R ), radially symmetric, J ≥ 0 with ∫ J = 1. Our assumption on the initial d...

Large time behavior of nonlocal aggregation models with nonlinear diffusion

The aim of this paper is to establish rigorous results on the large time behavior of nonlocal models for aggregation, including the possible presence of nonlinear diffusion terms modeling local repulsions. We show that, as expected from the practical motivation as well as from numerical simulations, one obtains concentrated densities (Dirac δ distributions) as stationary solutions and large tim...

A Nonlocal Quenching Problem Arising in Micro-electro Mechanical System

In this paper, we study a nonlocal parabolic problem arising in the study of microelectro mechanical system. The nonlocal nonlinearity involved is related to an integral over the spatial domain. We first give the structure of stationary solutions. Then we derive the convergence of global (in time) solution to the maximal solution as the time tends to infinity. Finally, we provide some quenching...

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 bo...