Asymptotic behavior for nonlocal diffusion equations

✍ Scribed by Emmanuel Chasseigne; Manuela Chaves; Julio D. Rossi

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 220 KB
• Volume: 86
• Category: Article
• ISSN: 0021-7824
• DOI: 10.1016/j.matpur.2006.04.005

✦ Synopsis

We study the asymptotic behavior for nonlocal diffusion models of the form u t = J * uu in the whole R N or in a bounded smooth domain with Dirichlet or Neumann boundary conditions. In R N we obtain that the long time behavior of the solutions is determined by the behavior of the Fourier transform of J near the origin, which is linked to the behavior of J at infinity.

, the asymptotic behavior is the same as the one for solutions of the evolution given by the α/2 fractional power of the Laplacian. In particular when the nonlocal diffusion is given by a compactly supported kernel the asymptotic behavior is the same as the one for the heat equation, which is yet a local model. Concerning the Dirichlet problem for the nonlocal model we prove that the asymptotic behavior is given by an exponential decay to zero at a rate given by the first eigenvalue of an associated eigenvalue problem with profile an eigenfunction of the first eigenvalue. Finally, we analyze the Neumann problem and find an exponential convergence to the mean value of the initial condition.