2015/04/27 by Cortázar, Carmen, Elgueta, Manuel, Quirós, Fernando +1
#35R09 #45K05 #45M05 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1504.07301
We study the long time behavior of bounded, integrable solutions to a nonlocal diffusion equation, ∂ t u=J*u-u, where J is a smooth, radially symmetric kernel with support Bd(0)⊂ℝ2. The problem is set in an exterior two-dimensional domain which excludes a hole H, and with zero Dirichlet data on H. In the far field scale, ξ1≤ |x|t-1/2≤ ξ2 with ξ1,ξ2>0, the scaled function log t u(x,t) behaves as a multiple of the fundamental solution for the local heat equation with a certain diffusivity determined by J. The proportionality constant, which characterizes the first non-trivial term in the asymptotic behavior of the mass, is given by means of the asymptotic \lq logarithmic momentum' of the solution, limt→∞∫ℝ2u(x,t)log|x| dx. This asymptotic quantity can be easily computed in terms of the initial data. In the near field scale, |x|≤ t1/2h(t) with limt→∞ h(t)=0, the scaled function t(log t)2u(x,t)/log |x| converges to a multiple of ϕ(x)/log |x|, where ϕ is the unique stationary solution of the problem that behaves as log|x| when |x|→∞. The proportionality constant is obtained through a matching procedure with the far field limit. Finally, in the very far field, |x|≥ t1/2 g(t) with g(t)→∞, the solution is proved to be of order o((tlog t)-1).