The Dirichlet problem for some nonlocal diffusion equations

Abstract : We study the Dirichlet problem for the non-local diffusion equation $u_t=\int\{u(x+z,t)-u(x,t)\}\dmu(z)$, where $\mu$ is a $L^1$ function and $u=\varphi$ on $\partial\Omega\times(0,\infty)$'' has to be understood in a non-classical sense. We prove existence and uniqueness results of solutions in this setting. 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.
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https://hal.archives-ouvertes.fr/hal-00132526
Contributor : Emmanuel Chasseigne <>
Submitted on : Wednesday, May 30, 2007 - 3:25:54 PM
Last modification on : Friday, February 15, 2019 - 1:22:22 AM
Document(s) archivé(s) le : Tuesday, September 21, 2010 - 12:36:23 PM

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Emmanuel Chasseigne. The Dirichlet problem for some nonlocal diffusion equations. Differential and integral equations, Khayyam Publishing, 2007, 20, pp.1389-1404. ⟨hal-00132526v2⟩

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