Nonlocal higher order evolution equations

In this paper we study the asymptotic behavior of solutions to the nonlocal operator ut(x, t) = (−1) n−1 (J ∗ Id − 1)n (u(x, t)), x ∈ R which is the nonlocal analogous to the higher order local evolution equation vt = (−1)(∆)v. We prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity. Moreover, we prove that the solutions of the nonlocal problem converge to the solution of the higher order problem with right hand side given by powers of the Laplacian when the kernel J is rescaled in an appropriate way.