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 ∈ ℝ ^N , which is the nonlocal analogous to the higher order local evolution equation v _t = (−1) ⁿ⁻¹(Δ) 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.

Keywords:

• nonlocal diffusion
• higher order
• asymptotic behaviour

AMS Subject Classifications:

• Primary 35B40
• Secondary 45A07
• 45G10

Acknowledgements

C.-B. Schönlieb is partially supported by the DFG Graduiertenkolleg 1023 Identification in Mathematical Models: Synergy of Stochastic and Numerical Methods, by the project WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the Visual Arts project No. CI06 003 and by the FFG project Erarbeitung neuer Algorithmen zum Image Inpainting project No. 813610. Further, this publication is based on the work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). J.D. Rossi is partially supported by UBA X066, CONICET (Argentina) and SIMUMAT (Spain).