2013/08/22 by Cortazar, Carmen, Elgueta, Manuel, Quiros, Fernando +1
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1308.4897
We study the large time behavior of solutions to a non-local diffusion equation, ut=J*u-u with J smooth, radially symmetric and compactly supported, posed in ℝ+ with zero Dirichlet boundary conditions. In sets of the form x≥ ξt1/2, ξ>0, the outer region, the asymptotic behavior is given by a multiple of the dipole solution for the local heat equation, and the solution is O(t-1). The proportionality constant is determined from a conservation law, related to the asymptotic first momentum. On compact sets, the inner region, after scaling the solution by a factor t3/2, 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 outer behavior. Since the outer and the inner region do not overlap, the matching is quite involved. It has to be done for the scaled function t3/2u(x,t)/x, which takes into account that different scales lead to different decay rates.