Diffusion in an annihilating environment

Abstract

In this paper we study the following system of reaction–diffusion equations: ∂ ϱ / ∂ t = Δ ϱ - V ϱ + λ δ 0 , ϱ ( 0 , x ) ≡ 0 , ∂ V / ∂ t = - ϱ V , V ( 0 , x ) ≡ 1 . Here ϱ ( t , x ) and V ( t , x ) are functions of time t ∈ [ 0 , ∞ ) and space x ∈ R d . This system describes a continuum version of a model in which particles are injected at the origin at rate λ , perform independent simple symmetric random walks on Z d , and are annihilated at rate 1 by traps located at the sites of Z d in such a way that the trap disappears with the particle. This lattice model was studied by a number of authors, who obtained the asymptotic size and shape of the front separating the zone of particles from the zone of traps as well as the asymptotic particle density profile to leading order, in the limit of large time. The continuum model has similar behavior but allows for a more detailed study. As t increases, the particle density ϱ ( t , · ) inflates and the trap density V ( t , · ) deflates on a growing ball with radius R * ( t ) centered at the origin. We derive the sharp asymptotics of the front position R * ( t ) , identify the shape of V ( t , · ) near the surface of the ball, and obtain the limiting profile of ϱ ( t , · ) inside the ball after appropriate scaling. We also identify the analogues of the total number and the age distribution of particles that are alive. It turns out that the cases d ⩾ 3 , d = 2 , and 1 exhibit different behavior.