The Dirichlet problem for some nonlocal diffusion equations. (English) Zbl 1211.47088

Chasseigne, Emmanuel

Differ. Integral Equ. 20, No. 12, 1389-1404 (2007).

Summary: We study the Dirichlet problem for the non-local diffusion equation \(u_t=\int [u(x+z,t)-u(x,t)]\,d\mu (z)\), 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.

MSC:

47G20	Integro-differential operators
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B50	Maximum principles in context of PDEs
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

non-local diffusion equation; Dirichlet problem; existence; uniqueness