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2007 The Dirichlet problem for some nonlocal diffusion equations
Emmanuel Chasseigne
Differential Integral Equations 20(12): 1389-1404 (2007).

Abstract

We study the Dirichlet problem for the non-local diffusion equation $u_t=\int\{u(x+z,t)-u(x,t)\}{\,\mathrm{d}\mu}(z)$, 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.

Citation

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Emmanuel Chasseigne. "The Dirichlet problem for some nonlocal diffusion equations." Differential Integral Equations 20 (12) 1389 - 1404, 2007.

Information

Published: 2007
First available in Project Euclid: 20 December 2012

zbMATH: 1211.47088
MathSciNet: MR2377023

Subjects:
Primary: 35K20
Secondary: 35B05 , 35D10

Rights: Copyright © 2007 Khayyam Publishing, Inc.

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Vol.20 • No. 12 • 2007
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