2014/12/01 by Carmen Cortázar, Cortázar, Carmen, Manuel Elgueta +5
Computer Science · Mathematics · #35R09 #45K05 #45M05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1412.0731
openalex publication_date 2014/12/01 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We study the long time behavior of solutions to the nonlocal diffusion\nequation \∂t u=J*u-u in an exterior one-dimensional domain, with zero\nDirichlet data on the complement. In the far field scale,\n\ξ1\≤|x|t-1/2\≤\ξ2, \ξ1,\ξ2>0, this behavior is given by a\nmultiple of the dipole solution for the local heat equation with a diffusivity\ndetermined by J. However, the proportionality constant is not the same on\n\ℝ+ and \ℝ-: it is given by the asymptotic first momentum\nof the solution on the corresponding half line, which can be computed in terms\nof the initial data. In the near field scale, |x|\≤ t1/2h(t),\n\limt\→\∞h(t)=0, the solution scaled by a factor t3/2/(|x|+1)\nconverges to a stationary solution of the problem that behaves as b^\±x as\nx\→\±\∞. The constants b^\± are obtained through a matching\nprocedure with the far field limit. In the very far field, |x|\≥t1/2\ng(t), g(t)\→\∞, the solution has order o(t-1).\n