TY - JOUR
T1 - Nonlocal higher order evolution equations
AU - Rossi, Julio D.
AU - Schönlieb, Carola-Bibiane
N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: C.-B. Schonlieb 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).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.
PY - 2010/6
Y1 - 2010/6
N2 - In this article, we study the asymptotic behaviour of solutions to the nonlocal operator ut(x, t)1/4(-1)n-1 (J*Id -1)n (u(x, t)), x ∈ ℝN, which is the nonlocal analogous to the higher order local evolution equation vt(-1)n-1(Δ)nv. 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. © 2010 Taylor & Francis.
AB - In this article, we study the asymptotic behaviour of solutions to the nonlocal operator ut(x, t)1/4(-1)n-1 (J*Id -1)n (u(x, t)), x ∈ ℝN, which is the nonlocal analogous to the higher order local evolution equation vt(-1)n-1(Δ)nv. 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. © 2010 Taylor & Francis.
UR - http://hdl.handle.net/10754/598997
UR - http://www.tandfonline.com/doi/abs/10.1080/00036811003735824
UR - http://www.scopus.com/inward/record.url?scp=77952731075&partnerID=8YFLogxK
U2 - 10.1080/00036811003735824
DO - 10.1080/00036811003735824
M3 - Article
SN - 0003-6811
VL - 89
SP - 949
EP - 960
JO - Applicable Analysis
JF - Applicable Analysis
IS - 6
ER -