RANDOM WALKS AND THE POROUS MEDIUM EQUATION
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Date
2009
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Publisher
UNION MATEMATICA ARGENTINA
Abstract
In this paper we consider a fixed initial condition u(0) and we study the limit as epsilon -> 0 of u(epsilon), the solution to the re-scaled problem
u(t)(x,y) = 1/epsilon(2)(integral(R)J(x-y/epsilon u(y,t))dy/epsilon-u(x,t)) in R x [0, infinity)
with initial condition u(epsilon)(x, 0) = u(0)(x). Here J is a smooth non negative even function supported in the interval [-1, 1]. Moreover it is assumed that integral J - 1 and that J is decreasing on [0, 1].
We prove that, under adequate hypothesis on the initial condition, the limit
lim(epsilon -> 0) u(epsilon) = u
is the solution to the well known porous medium equation v(t) = D(v(3))(xx), with initial condition u(epsilon)(x, 0) - u(0)(x) for a suitable constant D.
u(t)(x,y) = 1/epsilon(2)(integral(R)J(x-y/epsilon u(y,t))dy/epsilon-u(x,t)) in R x [0, infinity)
with initial condition u(epsilon)(x, 0) = u(0)(x). Here J is a smooth non negative even function supported in the interval [-1, 1]. Moreover it is assumed that integral J - 1 and that J is decreasing on [0, 1].
We prove that, under adequate hypothesis on the initial condition, the limit
lim(epsilon -> 0) u(epsilon) = u
is the solution to the well known porous medium equation v(t) = D(v(3))(xx), with initial condition u(epsilon)(x, 0) - u(0)(x) for a suitable constant D.
Description
Keywords
Nonlocal diffusion, free boundaries, NONLOCAL DIFFUSION-PROBLEMS, BOUNDARY-CONDITIONS, HEAT-EQUATION, APPROXIMATE