The Dirichlet problem for some nonlocal diffusion equations - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2007

The Dirichlet problem for some nonlocal diffusion equations

Résumé

We study an ad-hoc notion of Dirichlet problem for the non-local diffusion equation $u_t=\int\{u(x+z,t)-u(x,t)\}\dmu(z)$, where $``u=\varphi$ on $\partial\Omega\times(0,\infty)$'' has to be understood in a non-classical sense. We then prove existence and uniqueness results of solutions in this setting when $\mu$ is a $L^1$ function. Moreover, we prove that our solutions coincide with those obtained through the standard ``vanishing viscosity method'', but show that a boundary layer occurs: the solution does not take the boundary data in the classical sense on $\partial\Omega$, a phenomenon related to the non-local character of the equation. Finally, we show that in a bounded domain, some regularization may occur, contrary to what happens in the whole space.
Fichier principal
Vignette du fichier
DirNonLocal.pdf (186.68 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-00132526 , version 1 (21-02-2007)
hal-00132526 , version 2 (30-05-2007)

Identifiants

Citer

Emmanuel Chasseigne. The Dirichlet problem for some nonlocal diffusion equations. 2007. ⟨hal-00132526v1⟩
136 Consultations
135 Téléchargements

Altmetric

Partager

More