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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
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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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