Nonlocal higher order evolution equations

Abstract

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