In this paper, we employ the renormalization group method to study the long-time asymptotics of solutions to a class of nonlinear integral equations with a generalized heat kernel. The nonlinearities are classified and studied according to its role in the asymptotic behavior. Here we prove that the behavior, in the limit as t goes to infinity, remains unchanged when compared with the one in the linear case if the nonlinearities are the ones classified as irrelevant in the renormalization group sense.

1.
M.
Avellaneda
, “
Homogenization and renormalization: The mathematics of multi-scale random media and turbulent diffusion
,” in
Dynamical Systems and Probabilistic Methods in Partial Differential Equations
, Lectures in Applied Mathematics (
American Mathematical Society
,
1996
), Vol. 31, pp.
251
268
.
2.
G.
Braga
,
F.
Furtado
,
J.
Moreira
, and
L.
Rolla
, “
Renormalization group analysis of nonlinear diffusion equations with time dependent coefficients: Analytical results
,”
Discrete Contin. Dyn. Syst., Ser. B
7
,
699
715
(
2007
).
3.
G.
Braga
and
J.
Moreira
, “
Renormalization group analysis of nonlinear diffusion equations with time dependent coefficients and marginal perturbations
,”
J. Stat. Phys.
148
(
2
),
280
295
(
2012
).
4.
G.
Braga
,
F.
Furtado
, and
L.
Lee
, “
Numerical renormalization group algorithms for self-similar solutions of partial differential equations
,” e-print arXiv:1707.05544 (
2017
).
5.
G.
Braga
and
W.
Conti
, “
A multiscale asymptotic analysis of time evolution equations on the complex plane
,”
J. Math. Phys.
57
,
073504
(
2016
).
6.
J.
Bricmont
,
A.
Kupiainen
, and
G.
Lin
, “
Renormalization group and asymptotics of solutions of nonlinear parabolic equations
,”
Commun. Pure Appl. Math.
47
,
893
922
(
1994
).
7.
J.
Bricmont
and
A.
Kupiainen
, “
Renormalizing partial differential equations
,” In
Constructive Physics Results in Field Theory, Statistical Mechanics and Condensed Matter Physics. Lecture Notes in Physics
, edited by V. Rivasseau, Springer, Berlin, Heidelberg (1995), Vol. 446.
8.
L.
Chen
and
N.
Goldenfeld
, “
Numerical renormalization group calculations for similarity solutions and travelling waves
,”
Phys. Rev. E
51
,
5577
5581
(
1995
).
9.
G.
Dagan
, “
Theory of solute transport by groundwater
,”
Annu. Rev. Fluid Mech.
19
,
183
215
(
1987
).
10.
R. E.
Davis
, “
Lagrangian ocean studies
,”
Annu. Rev. Fluid Mech.
23
,
45
64
(
1991
).
11.
M.
Gell-Mann
and
F. E.
Low
, “
Quantum electrodynamics at small distances
,”
Phys. Rev.
95
,
1300
1312
(
1954
).
12.
J.
Glimm
,
W. B.
Lindquist
,
F.
Pereira
, and
Q.
Zhang
, “
A theory of macrodispersion for the scale-up problem
,”
Transp. Porous Media
13
,
97
122
(
1993
).
13.
N.
Goldenfeld
,
Lectures on Phase Transition and the Renormalization Group
(
Addison-Wesley
,
Reading
,
1992
).
14.
V.
Isaia
, “
Numerical simulation of universal finite time behavior for parabolic IVP via geometric renormalization group
,”
Discrete Contin. Dyn. Syst., Ser. B
22
(
9
),
3459
3481
(
2017
).
15.
K.
Ishige
,
T.
Kawakami
, and
K.
Kobayashi
, “
Asymptotics for a nonlinear integral equation with a generalized heat kernel
,”
J. Evol. Equations
14
,
749
777
(
2014
).
16.
K.
Ishige
,
T.
Kawakami
, and
K.
Kobayashi
, “
Global solutions for a nonlinear integral equation with a generalized heat kernel
,”
Discrete Contin. Dyn. Syst., Ser. S
7
,
767
783
(
2014
).
17.
L.
Lake
and
H.
Carroll
,
Reservoir Characterization
(
Academic Press
,
New York
,
1986
).
18.
A. J.
Majda
and
P. R.
Kramer
, “
Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena
,”
Phys. Rep.
314
,
237
574
(
1999
).
19.
H. K.
Moffatt
, “
Transport effects associated with turbulence with particular attention to the influence of helicity
,”
Rep. Prog. Phys
46
,
621
(
1983
).
20.
B.
Vollmayr-Lee
,
J.
Hanson
,
S.
McIsaac
, and
J.
Hellerick
, “
Renormalization group calculation of anomalous dimension in the trapping reaction
,”
Bull. Am. Phys. Soc.
62
,
4
(
2017
);
B.
Vollmayr-Lee
,
J.
Hanson
,
S.
McIsaac
, and
J.
Hellerick
,“
Anomalous dimension in a two-species reaction-diffusion system
,”
J. Phys. A: Math. Theor.
51
,
034002
(
2017
).
21.
K.
Wilson
, “
Renormalization group and critical phenomena. I
,”
Phys. Rev. B
4
,
3174
3183
(
1971
)
K.
Wilson
, “
Renormalization group and critical phenomena II
,”
Phys. Rev. B
4
,
3184
3205
(
1971
).
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