The Dirichlet problem for some nonlocal diffusion equations

Emmanuel Chasseigne (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

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.

Mots clés: general nonlocal operators; integro-differential equations; Dirichlet problem; Lévy operators; general nonlocal operators.

Domaines: Equations aux dérivées partielles [math.AP]

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