@misc{10481/56086,
year = {2018},
url = {http://hdl.handle.net/10481/56086},
abstract = {We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations:
L
u
(
x
)
:=
∫
R
N
K
(
x
,
y
)
(
u
(
y
)
−
u
(
x
)
)
d
y
,
where 
L
 acts on a real function 
u
 defined on 
R
N
, and we assume that 
K
(
x
,
y
)
 is uniformly strictly positive in a neighbourhood of 
x
=
y
. The inequality is a nonlocal analogue of the Nash inequality, and plays a similar role in the study of the asymptotic decay of solutions to the nonlocal diffusion equation 
∂
t
u
=
L
u
 as the Nash inequality does for the heat equation. The inequality allows us to give a precise decay rate of the 
L
p
 norms of 
u
 and its derivatives. As compared to existing decay results in the literature, our proof is perhaps simpler and gives new results in some cases.},
organization = {J. A. Cañizo was supported by the Spanish Ministerio de Economía y Competitividad
and the European Regional Development Fund (ERDF/FEDER), project
MTM2014-52056-P. A. Molino was partially supported by MINECO - FEDER
GrantMTM2015-68210-P(Spain), Junta de Andalucía FQM-116 (Spain) andMINECO
Grant BES-2013-066595 (Spain).},
publisher = {American Institute of Mathematical Sciences (AIMS)},
keywords = {Nonlocal diffusion},
keywords = {Entropy methods},
keywords = {Energy methods},
keywords = {Asymptotic behaviour},
title = {Improved energy methods for nonlocal diffusion problems},
author = {Cañizo Rincón, José Alfredo and Molino Salas, Alexis},
}