Asymptotic behavior for a nonlocal diffusion equation on the half line

Abstract

We study the large time behavior of solutions to a nonlocal diffusion equation, $u_t=J*u-u$ with $J$ smooth, radially symmetric and compactly supported, posed in $\mathbb{R}_+$ with zero Dirichlet boundary conditions. In the far-field scale, $\xi_1\le xt^{-1/2}\le \xi_2$ with $\xi_1,\xi_2>0$, the asymptotic behavior is given by a multiple of the dipole solution for the local heat equation, hence $tu(x,t)$ is bounded above and below by positive constants in this region for large times. The proportionality constant is determined from a conservation law, related to the asymptotic first momentum. In compact sets, after scaling the solution by a factor $t^{3/2}$, it converges to a multiple of the unique stationary solution of the problem that behaves as $x$ at infinity. The precise proportionality factor is obtained through a matching procedure with the far-field limit. Finally, in the very far-field, $x\ge t^{1/2} g(t)$ with $g(t)\to\infty$, the solution is proved to be of order $o(t^{-1})$.