The Dirichlet problem for some nonlocal diffusion equations

Abstract

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