A Nonlocal Diffusion Equation whose Solutions Develop a Free Boundary

Let \(J:\mathbb{R} \to \mathbb{R}\) be a nonnegative, smooth compactly supported function such that \(\int_\mathbb{R} {J(r)dr = 1.} \) We consider the nonlocal diffusion problem

with a nonnegative initial condition. Under suitable hypotheses we prove existence, uniqueness, as well as the validity of a comparison principle for solutions of this problem. Moreover we show that if u(·, 0) is bounded and compactly supported, then u(·, t) is compactly supported for all positive times t. This implies the existence of a free boundary, analog to the corresponding one for the porous media equation, for this model.

Authors and Affiliations

1. Departamento de Matemática, Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

   Carmen Cortazar, Manuel Elgueta & Julio D. Rossi

Corresponding author

Correspondence to Carmen Cortazar.

Communicated by Rafael D. Benguria

submitted 29/01/04, accepted 09/09/04

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