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Preface: Special issue on dissipative systems and applications with emphasis on nonlocal or nonlinear diffusion problems
April  2015, 35(4): 1391-1407. doi: 10.3934/dcds.2015.35.1391

## Asymptotic behavior for a nonlocal diffusion equation on the half line

 1 Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile, Chile 2 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049-Madrid, Spain, Spain

Received  June 2013 Revised  March 2014 Published  November 2014

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 $o(t^{-1})$.
Citation: Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391
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