Yamamoto,  Masakazu

EPrint Series of Department of Mathematics, Hokkaido University

ASYMPTOTIC EXPANSION OF SOLUTIONS TO THE DISSIPATIVE
EQUATION WITH ANOMALOUS DIFFUSION
Masakazu Yamamoto1
1. Introduction
The following Cauchy problem for the linear dissipative equation is studied by many authors:
(1.1)
½
@tu¡¢u+ a(t; x)u = 0; t > 0; x 2 Rn;
u(0; x) = u0(x); x 2 Rn;
where n 2 N and the coe±cient a : (0;1)£Rn ! R and the initial data u0 : Rn ! R are given
functions. Upon the suitable condition for a(t; x) and u0(x), the well-posedness, the global in
time existence and the decay of solutions to (1.1) are shown. Moreover the asymptotic expansion
of the solution to (1.1) as t!1 is derived (cf [6]). Here we consider those problems when the
dissipative e®ect on the equation is provided by \the anomalous di®usion". In this manuscript,
we de¯ne the Fourier transform and the Fourier inverse transform by
F ['](») := (2¼)¡n=2
Z
Rn
e¡ix¢»'(x)dx and F¡1['](x) := (2¼)¡n=2
Z
Rn
eix¢»'(»)d»:
Then, for µ > 0, the fractional Laplacian is given by
(¡¢)µ=2'(x) = F¡1
h
j»jµF [']
i
(x):
The fractional Laplacian with µ = 2 is the positive Laplacian. On the other hand, when
1 < µ < 2, this operator provides the anomalous di®usion on dissipative equations (see [2, 8]).
Namely, for the fundamental solution of @tu+ (¡¢)µ=2u = 0, we see the following property.
Lemma 1.1 ([1]). Let n 2 Z; µ > 0, Cµ := µ2µ¡1¼¡n2¡1 sin µ¼2 ¡
¡
n+µ
2
¢
¡
¡
µ
2
¢
and
(1.2) Gµ(t; x) := F¡1[e¡tj»jµ ](x):
Then the following property holds:
jxjn+µGµ(t; x)! Cµt as jxj ! 1
for any t > 0.
Here we remark that Gµ(t; x) is the fundamental solution of @tu+(¡¢)µ=2u = 0. When µ = 2,
the fundamental solution of @tu¡¢u = 0 is given by the Gaussian G(t; x) = (4¼t)¡n=2e¡jxj2=(4t).
This satis¯es
jxjMG(t; x)! 0 as jxj ! 1
for any t > 0 and M > 0. This property and Lemma 1.1 are not contradictory. Indeed, when
µ = 2, we see that Cµ = 0 in Lemma 1.1. When 1 < µ < 2, Lemma 1.1 immediately gives that
kx®Gµ(t)k1 = +1
1Graduate School of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan.
E-mail: yamamoto@cc.hirosaki-u.ac.jp
1
for any t > 0 and ® 2 Zn+ with j®j ¸ 2. We study the following linear dissipative equation with
anomalous di®usion:
(1.3)
½
@tu+ (¡¢)µ=2u+ a(t; x)u = 0; t > 0; x 2 Rn;
u(0; x) = u0(x); x 2 Rn;
where n 2 N; 1 < µ · 2. When µ = 2, this and (1.1) are equivalent. We assume the following
assumption for the coe±cient:
(1.4) kx®a(t)kp · C(1 + t)¡¹+
j®j
µ
+ n
µp
for any ® 2 Zn+ with j®j < º; n=(º ¡ j®j) < p · 1 and t > 0, where C is a positive constant,
and ¹ > 1 + 1=µ and º > 0 are some parameters. For example, if a(t; x) satis¯es the inequality
ja(t; x)j · C(1 + t)¡¹
D
(1 + t)¡1=µx
E¡º
for some C > 0 and any (t; x) 2 (0;1) £ Rn, then the condition (1.4) is satis¯ed. Here we
have used the notation hxi :=p1 + jxj2. We assume that the solution to (1.3) is well-posed in
Lp(Rn) with some 1 · p · 1 and satis¯es
(1.5) kx®u(t)kp · C(1 + t)¡
n
µ
(1¡ 1
p
)+
j®j
µ
for any t > 0; ® 2 Zn+ with j®j < n + µ and n=(n + µ ¡ j®j) < p · 1. Indeed, for some
problems, we obtain this inequality. We should remark that the anomalous di®usion causes
kx®u(t)k1 = +1 when j®j ¸ 2. Namely a moment of the solution with high-order diverges to
in¯nity. We consider the large-time behavior of the solution to (1.3). Especially we give the
estimate on the di®erence between the solution and its asymptotic expansion as t ! 1. For
(1.1) (namely, for (1.3) with µ = 2), Ishige, Ishiwata and Kawakami [6] derived the large-time
behavior of the solution completely. In their asymptotic expansion, the coe±cients contain the
moments of the solution. Unfortunately we cannot extend this idea to our problem since the
moments of the solution cannot be de¯ned. We provide the other way to reach our goal. For
some (l; ¯) 2 Z+ £ Zn+, we introduce the following notation:
Ul;¯(t; x) :=
@ltr¯Gµ(1 + t; x)
l!¯!
µZ
Rn
(¡1)l(¡y)¯u0(y)dy
¡
Z t
0
Z
Rn
(¡1¡ s)l(¡y)¯(au)(s; y)dyds
¶
;
(1.6)
where Gµ(t) is de¯ned by (1.2). Then we see the following proposition.
Proposition 1.2. Let n 2 N; 1 < µ · 2; ¹ > 1+1=µ and º > 0. Let (1.4) and (1.5) be satis¯ed.
Assume that u0 2 L1N (Rn) \ L1(Rn) for N := maxfm 2 Z+ j m < (¹¡ 1)µ; º + µg and u(t) is
the solution to (1.3), where
(1.7) L1N (Rn) :=
©
' 2 L1(Rn) ¯¯ jxjN' 2 L1(Rn)ª :
Let
(1.8) ¹¤N :=
©
(l; ¯) 2 Z+ £ Zn+
¯¯
µl + j¯j · Nª
2
and the functions fUl;¯(t)g(l;¯)2¹¤N be given by (1.6). Then the following estimate holds:°°°°u(t)¡ X
(l;¯)2¹¤N
Ul;¯(t)
°°°°
p
= o
³
t
¡n
µ
(1¡ 1
p
)¡N
µ
´
as t!1
for 1 · p · 1.
When µ = 2, the asymptotic expansion of solution of this type was ¯rstly observed by Es-
cobedo and Zuazua. In [4], they derived the asymptotic expansion of solutions to the heat-
convection equation. For the solution of the Navier-Stokes equation, the asymptotic expansion
was provided by Carpio [3], and Fujigaki and Miyakawa [5]. The large-time behavior of the so-
lution to the Keller-Segel equation in Lp(Rn) was considered by Nagai, Syukuinn and Umesako
[9], Kato [7], and Nagai and Yamada [10].
Since the conditions (1.4), (1.5) and N < minf(¹ ¡ 1)µ; º + µg are assumed, the coe±cientR1
0
R
Rn(¡1 ¡ s)l(¡y)¯(au)(s; y)dyds is uniformly integrable when (l; ¯) 2 ¹¤N holds. Hence,
under this assumption, the asymptotic expansion (1.6) is well-de¯ned. However, when N ¸
(¹ ¡ 1)µ or N ¸ º + µ is satis¯ed, some coe±cients in the asymptotic expansion diverge to
in¯nity. Thus, in this case, we cannot de¯ne the higher-order asymptotic expansion by the form
as (1.6). Before proceeding next step, we study the decay-rates of the solution as jxj ! 1.
Then we obtain the following proposition.
Proposition 1.3. Let n 2 N; 1 < µ · 2; ¹ > 1+1=µ and º > 0. Let (1.4) and (1.5) be satis¯ed.
Assume that u0 2 L1N (Rn) \ L1(Rn) for N := maxfm 2 Z+ j m < (¹¡ 1)µ; º + µg and u(t) is
the solution to (1.3), where L1N (Rn) is de¯ned by (1.7). Let
(1.9) ¤N :=
©
(l; ¯) 2 Z+ £ Zn+
¯¯
µl + j¯j < Nª
and
(1.10) U¤N (t; x) :=
X
(l;¯)2¤N
Ul;¯(t; x);
where the functions fUl;¯(t)g(l;¯)2¤N be de¯ned by (1.6). Then u(t) ¡ U¤N (t) 2 L1N (Rn) holds
for any t > 0. Moreover there exists a positive constant C such that
kx® (u(t)¡ U¤N (t))k1 · C(1 + t)¡
N¡j®j
µ
for any t > 0 and ® 2 Zn+ with j®j · N .
We should remark that, for any ® 2 Zn+ with j®j ¸ 2, the function U¤N (t) in Proposition
1.3 satis¯es kx®U¤N (t)k1 = 1. Thus this function gives the approximation of the solution as
jxj ! 1. Proposition 1.2 and 1.3 state that the asymptotic expansion of the solution as t!1
and the approximation of the solution as jxj ! 1 are given by the same form. We derive the
higher-order asymptotic expansion of the solution by employing this proposition.
Notation. Throughout this manuscript, we use the following notation. For any x = (x1; : : : ; xn)
and y = (y1; : : : ; yn) 2 Rn, we denote x ¢ y :=
Pn
j=1 xjyj ; jxj2 := x ¢ x and hxi :=
p
1 + jxj2. For
1 · p · 1 and µ > 0, Lp(Rn) denotes the Lebesgue spaces and W µ;p(Rn) denotes the Sobolev
spaces. The norm of Lp(Rn) is represented by k ¢ kp. For s > 0, we de¯ne the weighted L1(Rn)
space by L1s(Rn) := f' 2 L1(Rn) j k'kL1s :=
R
Rnhxisj'(x)jdx <1g. For f = f(x) and g = g(x),
3
we denote the convolution by f ¤ g(x) := RRn f(x¡ y)g(y)dy. The gamma function ¡ = ¡(p) for
p > 0 is provided by ¡(p) :=
R1
0 e
¡ttp¡1dt. Varias constants are simply denoted by C.
2. Preliminaries
Before stating our results, we study some basic properties of Gµ(t) := F¡1[e¡tj»jµ ].
Lemma 2.1. For any l 2 Z+ and ® 2 Zn+, there exists a positive constant C > 0 such that¯¯¯
@ltr®Gµ(t; x)
¯¯¯
· Ct¡nµ¡l¡ j®jµ
D
t¡1=µx
E¡n¡µ¡µl¡j®j
for any (t; x) 2 (0;1)£ Rn.
Proof. This lemma is proved by employing [11, Theorem 3.1.]. Here we omit the proof. ¤
By applying Taylor's formula, Lemma 2.1 and Hausdorf-Young's inequality, we obtain the
following lemma.
Lemma 2.2. Let N 2 Z+ and u0 2 L1N (Rn). Then the following estimate holds for any
1 · p · 1:°°°°Gµ(t) ¤ u0 ¡ X
(l;¯)2¹¤N
@ltr¯Gµ(1 + t)
l!¯!
Z
Rn
(¡1)l(¡y)¯u0(y)dy
°°°°
p
= o
³
t
¡n
µ
(1¡ 1
p
)¡N
µ
´
as t!1, where ¹¤N is de¯ned by (1.8).
When we study the decay of Gµ(t) ¤ u0(x) as jxj ! 1, we obtain the approximation of this
by the same form as in Lemma 2.2.
Lemma 2.3. Let N 2 Z+ and u0 2 L1N (Rn). Then the following inequality holds for any t > 0
and ® 2 Zn+ with j®j · N :°°°°x®µGµ(t) ¤ u0 ¡ X
(l;¯)2¤N
@ltr¯Gµ(1 + t)
l!¯!
Z
Rn
(¡1)l(¡y)¯u0(y)dy
¶°°°°
1
·C(1 + t)¡N¡j®jµ ;
where C is a positive constant and ¤N is de¯ned by (1.9).
3. Main Results
In order to derive the asymptotic expansion of the solution, we introduce the following corre-
sponding integral equation:
(3.1) u(t) = Gµ(t) ¤ u0 ¡
Z t
0
Gµ(t¡ s) ¤ (au)(s)ds:
The solution to (3.1) is called the mild solution of (1.3). Generally speaking a mild solution
solves an original Cauchy problem if it has su±ciently high regularity. Hereafter we consider the
mild solution for deriving the asymptotic expansion of the solution of (1.3). For N = minfm 2
Z+ j m < (¹¡ 1)µ; º + µg, the nonlinear term on the right hand side of (3.1) is split asZ t
0
Gµ(t¡ s) ¤ (au)(s)ds =
X
(l;¯)2¤N
Jl;¯(t) +
Z t
0
Gµ(t¡ s) ¤ (a(u¡ U¤N )) (s)ds;
4
where
(3.2) Jl;¯(t; x) :=
Z t
0
Gµ(t¡ s) ¤ (aUl;¯)(s)ds
for (l; ¯) 2 ¤N and Ul;¯(t) is de¯ned by (1.6). We introduce the function ~Ul;¯ = ~Ul;¯(t; x) by
~Ul;¯(t; x) :=
@ltr¯Gµ(1 + t; x)
l!¯!
µZ
Rn
(¡1)l(¡y)¯u0(y)dy
¡
Z t
0
Z
Rn
(¡1¡ s)l(¡y)¯ (a(u¡ U¤N )) (s; y)dyds
¶(3.3)
for some (l; ¯) 2 Z+ £ Zn+, where U¤N (t) is de¯ned by (1.10). By employing the assumption
(1.4) and Proposition 1.3, we see that ~Ul;¯(t) is well-de¯ned when (l; ¯) 2 ¹¤2N holds, where ¹¤2N
is de¯ned as (1.8). Then we give our main result in the following theorems.
Theorem 3.1. Let n 2 N; 1 < µ · 2; ¹ > 1 + 1=µ and º > 0. Let (1.4) and (1.5) be satis¯ed.
Assume that u0 2 L12N (Rn) \ L1(Rn) for N := maxfm 2 Z+ j m < (¹ ¡ 1)µ; º + µg and u(t)
is the solution to (1.3), where L12N (Rn) is de¯ned as (1.7). Let ¤N and ¹¤2N be de¯ned as (1.8)
and (1.9), and the functions fJk;®(t)g(k;®)2¤N and f ~Ul;¯(t)g(l;¯)2¹¤2N be given by (3.2) and (3.3).
Then the following estimate holds:°°°°u(t)¡ X
(k;®)2¤N
Jk;®(t)¡
X
(l;¯)2¹¤2N
~Ul;¯(t)
°°°°
p
= o
³
t
¡n
µ
(1¡ 1
p
)¡ 2N
µ
´
as t!1
for 1 · p · 1.
Theorem 3.2. Let n 2 N; 1 < µ · 2; ¹ > 1 + 1=µ and º > 0. Let (1.4) and (1.5) be satis¯ed.
Assume that u0 2 L12N (Rn) \ L1(Rn) for N := maxfm 2 Z+ j m < (¹ ¡ 1)µ; º + µg and u(t)
is the solution to (1.3), where L12N (Rn) is de¯ned as (1.7). Let ¤N and ¤2N be de¯ned as (1.9)
and the functions fJk;®(t)g(k;®)2¤N and f ~Ul;¯(t)g(l;¯)2¤2N be given by (3.2) and (3.3). Then
u(t)¡
X
(k;®)2¤N
Jk;®(t)¡
X
(l;¯)2¤2N
~Ul;¯(t) 2 L12N (Rn)
holds for any t > 0. Moreover there exists a positive constant C such that:°°°°x®µu(t)¡ X
(k;®)2¤N
Jk;®(t)¡
X
(l;¯)2¤2N
~Ul;¯(t)
¶°°°°
1
· C(1 + t)¡ 2N¡j®jµ
for any t > 0 and ® 2 Zn+ with j®j · 2N .
For N := maxfm 2 Z+ j m < (¹ ¡ 1)µ; º + µg, Proposition 1.2 and 1.3 give the Nth-order
asymptotic expansion of the solution. Theorem 3.1 and 3.2 provide the 2Nth-order asymptotic
expansion. We prove those theorems by employing Proposition 1.2 and 1.3. Similarly, by
applying Theorem 3.1 and 3.2, we obtain a 3Nth-order asymptotic expansion. By repeating
this procedure, we can derive an mNth-order asymptotic expansion for arbitrary large m 2 Z+.
5
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Asymptotic expansion of solutions to the dissipative equation with anomalous diffusion

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