a suficient condition for the existence of periodic solution for a reation diffusion equation with infinite delay

Applied Mathematics and Computation 148 (2004) 453–460
www.elsevier.com/locate/amc

A sufficient condition for the existence
of periodic solution for a reaction
diffusion equation with infinite delay q
Yanbin Tang *, Li Zhou
Department of Mathematics, Huazhong University of Science and Technology, Wuhan,
Hubei 430074, PR China

Abstract
For a periodic reaction diffusion equation with infinite delay, a sufficient condition of
existence and uniqueness of periodic solution is given. By using the periodic monotone
method, the asymptotic behavior of the time-dependent solutions is investigated, that is,
the time-dependent solutions converge to the corresponding periodic solution in the
time variable.
Ó 2002 Elsevier Inc. All rights reserved.
Keywords: Reaction diffusion equation; Periodic solution; Existence; Asymptotic behavior; Delay

1. Introduction
In [1], Redlinger considered a single species population model with diffusion
and infinite delay

Luðt; xÞ ¼ uðt; xÞ a À buðt; xÞ À

Z

1


kðsÞuðt À s; xÞ ds ;

ðt; xÞ 2 Rþ Â X;

0

ð1:1Þ

q

The project supported by National Natural Science Foundation of China.
Corresponding author.
E-mail address: (Y. Tang).

*

0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved.
doi:10.1016/S0096-3003(02)00860-3


454

Y. Tang, L. Zhou / Appl. Math. Comput. 148 (2004) 453–460

ou
ðt; xÞ ¼ 0;
on

ðt; xÞ 2 Rþ Â oX;

uðt; xÞ ¼ /ðt; xÞ;

ð1:2Þ

ðt; xÞ 2 RÀ Â X;

ð1:3Þ

where L ¼ ðo=otÞ À M, a and b are positive constants, X is a bounded domain
in Rn with its boundary oX being a C 2 ––manifold, / 2 CðRÀ Â XÞ is a bounded
nonnegative function and /ð0; ÁÞ 2 C 1 ðXÞ. Redlinger obtained the following
result:
1.1. Suppose that k 2 CðRþ Þ \ L1 ðRþ Þ which k 6¼ 0 and b >
RTheorem
1
jkðsÞj
ds,
then the initial boundary value problem (1.1)–(1.3) has a unique
0
bounded, nonnegative and regular solution uðt; xÞ for each positive, continuous
and bounded function / on RÀ Â X. Moreover, if /ð0;R ÁÞ 6¼ 0, then uðt; xÞ > 0 for
1
all ðt; xÞ 2 ð0; þ1Þ Â X and limt!1 uðt; xÞ ¼ a=ðb þ 0 kðsÞ dsÞ.
In [2], Shi and Chen extended Theorem 1.1 to the case of a Volterra reaction
diffusion equation with variable coefficients as following:


Z 1
Luðt; xÞ ¼ uðt; xÞ aðt; xÞ À bðt; xÞuðt; xÞ À cðt; xÞ
uðt À s; xÞ dlðsÞ ;
0

ð1:4Þ
where a, b and c are sufficiently smooth functions and 0 < a1 6 aðt; xÞ 6 a2 ,
0 < b1 6 bðt; xÞ 6 b2 , 0 < c1 6 cðt; xÞ 6 c2 , lðÁÞ is of bounded variation with
lð0Þ ¼ 0.
Let MðtÞ denote the total variation of lðÁÞ on ½0; tŠ and M Æ ðtÞ ¼
½MðtÞ Æ lðtފ=2 for all t 2 Rþ , then MðtÞ and M Æ ðtÞ are nonnegative and nondecreasing on Rþ . Denote M0 ¼ limt!þ1 MðtÞ, M0Æ ¼ limt!þ1 M Æ ðtÞ, l0 ¼
limt!þ1 lðtÞ. Shi and Chen obtained the following asymptotic behavior:
Theorem 1.2. Suppose that b1 > c1 M0À , a1 ðb1 À c1 M0À Þ > c2 a2 M0þ and uðt; xÞ is a
solution of (1.2)–(1.4). Then there exist positive constants a, bð0 < a 6 bÞ such
that
a 6 lim inf min uðt; xÞ 6 lim sup max uðt; xÞ 6 b:
t!þ1

x2X

t!þ1

x2X

In [3], Zhou and Fu considered a periodic logistic delay equation with discrete delay effect as following:
Luðt; xÞ ¼ uðt; xÞ½1 þ ðt; xÞ À buðt; xÞ À cuðt À s; xފ;

ðt; xÞ 2 Rþ Â X;
ð1:5Þ

where ðt; xÞ is T -periodic in t and there is a constant ^ with 0 < ^ ( 1 such that
À^
 6 ðt; xÞ 6 ^ on X  ½0; T Š, b and c are positive constants. They obtained
the following conclusion.

Y. Tang, L. Zhou / Appl. Math. Comput. 148 (2004) 453–460

455

Theorem 1.3. Let b > c and ^ be sufficiently
small. ThenEthere is a unique T D
^
Ã
1
1

^
; bþc
þ bÀc
periodic solution u in the sector J ¼ bþc À bÀc
. Moreover, for any
initial function /ðt; xÞ 2 J , the corresponding solution uðt; xÞ of (1.5), (1.2), (1.3)
satisfies uðt; xÞ ! uà ðt; xÞ as t ! þ1 uniformly in x 2 X.
In this paper we consider the following periodic reaction diffusion equation
with infinite delay:


Z 1
Luðt; xÞ ¼ uðt; xÞ aðt; xÞ À bðt; xÞuðt; xÞ À cðt; xÞ
uðt À s; xÞ dlðsÞ ;
0

ð1:6Þ

ou
ðt; xÞ ¼ 0;
on

ðt; xÞ 2 Rþ Â oX;

uðt; xÞ ¼ /ðt; xÞ;

ð1:7Þ

ðt; xÞ 2 RÀ Â X:

ð1:8Þ

where a, b and c are sufficiently smooth functions, and strictly positive and
periodic in the time variable t with period T > 0.
Given a function g ¼ gðt; xÞ which is continuous, strictly positive and
T -periodic in t on R Â X. Let gM and gL denote the maximum and minimum of
g on R Â X respectively.
In this paper we consider the existence and uniqueness of the periodic
solution to the periodic boundary value problem (1.6) and (1.7), and the asymptotic behavior of the solution to the initial boundary value problem (1.6)–
(1.8).

2. Main result
Let a and b be given by the following system of linear equations:
aM À bL b À cL aM0þ þ cM bM0À ¼ 0;

ð2:1Þ

aL À bM a À cM bM0þ þ cL aM0À ¼ 0:

We assume that
aL bL > cM ðaL M0À þ aM M0þ Þ:

ð2:2Þ

Then it is easy to solve (2.1) and a unique solution is given by
a¼
b¼

aL ðbL À cM M0À Þ À aM cM M0þ
ðbL À cM M0À ÞðbM À cL M0À Þ À cL cM ðM0þ Þ
aM ðbM À cL M0À Þ À aL cL M0þ

2

ðbL À cM M0À ÞðbM À cL M0À Þ À cL cM ðM0þ Þ

;
ð2:3Þ

2

:


456

Y. Tang, L. Zhou / Appl. Math. Comput. 148 (2004) 453–460

Eq. (2.2) implies that 0 < a 6 b. We obtain the following theorem by making

use of the method of T -upper and lower solutions and the bootstrap technique
of periodic monotone iteration developed in [3–6].
Theorem 2.1. Assume that (2.2) holds true, and also
aM À 2bL a À cL aM0þ þ cM bM0þ þ 2cM bM0À < 0;

ð2:4Þ

then problem (1.6) and (1.7) has a unique T -periodic solution uà ðt; xÞ in the sector
ha; bi, and this solution is an attractor of system (1.6)–(1.8). That is, for any
initial function /ðt; xÞ 2 ha; bi, the corresponding solution uðt; xÞ of (1.6)–(1.8)
satisfies
uðt; xÞ ! uà ðt; xÞ;

t ! 1ðx 2 XÞ:

Proof. We know that the problem (1.6)–(1.8) is equivalent to the following:


Z 1
Z 1
þ
À
Luðt; xÞ ¼ u a À bu À c
uðt À s; xÞ dM ðsÞ þ c
uðt À s; xÞ dM ðsÞ ;
0

0

ð2:5Þ

ou
ðt; xÞ ¼ 0;
on

ðt; xÞ 2 Rþ Â oX;

uðt; xÞ ¼ /ðt; xÞ;

ð2:6Þ

ðt; xÞ 2 RÀ Â X:

ð2:7Þ

From (2.1), it is easy to check that b and a are the constant coupled upper
and lower solutions of (2.5) and (2.6). According to the bootstrap technique
developed in [4] and the existence theorem of quasi-solutions given in [3],
Theorem 3.2, there is a pair of T -periodic quasi-solutions hðt; xÞ and hðt; xÞ with
ð2:8Þ

a 6 hðt; xÞ 6 hðt; xÞ 6 b
such that



Z 1
Z 1
Lhðt; xÞ ¼ h a À bh À c
hðt À s; xÞ dM þ ðsÞ þ c

hðt À s; xÞ dM À ðsÞ ;
0
0


Z 1
Z 1
þ
hðt À s; xÞ dM ðsÞ þ c
hðt À s; xÞ dM À ðsÞ ;
Lhðt; xÞ ¼ h a À bh À c
0

0

þ

ðt; xÞ 2 R Â X;
oh
oh
ðt; xÞ ¼ ðt; xÞ ¼ 0;
on
on

ðt; xÞ 2 Rþ Â oX:
ð2:9Þ

Moreover, for any initial function /ðt; xÞ satisfying a 6 / 6 b in RÀ Â X, the
corresponding solution uðt; xÞ of (2.5)–(2.7) satisfies

Y. Tang, L. Zhou / Appl. Math. Comput. 148 (2004) 453–460

457

t ! 1ðx 2 XÞ:

hðt; xÞ 6 uðt; xÞ 6 hðt; xÞ;

ð2:10Þ

According to the equivalence of problem (2.5)–(2.7) and problem (1.6)–(1.8),
the solution of (1.6)–(1.8) also possesses the property (2.10).
From (2.9), we know that the T -periodic quasi-solution hðt; xÞ and hðt; xÞ of
(2.5) and (2.6) are not the solutions of the problem in general. It is obvious that
hðt; xÞ and hðt; xÞ is a pair of coupled T -periodic upper and lower solutions
of (1.6) and (1.7), namely,


Z 1
Lhðt; xÞ P h a À bh À c
hðt À s; xÞ dlðsÞ ;
0


Z
Lhðt; xÞ 6 h a À bh À c


hðt À s; xÞ dlðsÞ ;

1

ðt; xÞ 2 Rþ Â X;

ð2:11Þ

0

oh
oh
ðt; xÞ ¼ ðt; xÞ ¼ 0;
on
on

ðt; xÞ 2 Rþ Â oX:

Therefore, if hðt; xÞ ¼ hðt; xÞ, then (2.11) implies that the quasi-solution hðt; xÞ
(or hðt; xÞ) is exactly the T -periodic solution of (1.6) and (1.7).
We are going to show that hðt; xÞ ¼ hðt; xÞ, provided that (2.4) holds true.
Due to the periodicity of hðt; xÞ and hðt; xÞ, (2.9) implies that
0¼

Z

Z

Z T Z
o
ðh À hÞ dx ¼

dt ðh À hÞ
ot
X
X
0
0



Z 1
Z 1
Z
hs dM þ ðsÞ þ c
hs dM À ðsÞ dx À
 Mh þ h a À bh À c
T

dt

ðh À hÞ

0

Â

0



Z

ðh À hÞ Mh þ h a À bh À c

Z

¼À

Z

Z

T
0

À

jrðh À hÞj2 dx þ

dt
Z

Z

T

hs dM þ þ h

Z

0

6 ðaM À 2bL aÞ

Z 
dt
X

Z

T

Z

Z

T

1

hs dM À þ h

1
0

dx

X

Z

1

hs dM þ À h
0

2

Z



cðh À hÞ

dt

X

cðh À hÞ2

hs dM À ðsÞ

aðh À hÞ2 dx

ðh À hÞ dx

dt

1

X

0

Z
0

T

Z

0

1

Z
0

0

X


Z
 Àh

0

hs dM þ ðsÞ þ c

T

bðh þ hÞðh À hÞ2 dx þ

dt

Z

1

dt

X

0

À

Z

dt

0

0

X

T


hs dM þ dx

Z
0

1


hs dM À dx


458

Y. Tang, L. Zhou / Appl. Math. Comput. 148 (2004) 453–460

þ

Z

Z 

T

dt
0

þ

Z

Z 

T

dt
Z

chðh À hÞ

Z 
dt

0

1

hs dM
Z

À


dx

1

ðhs À hs Þ dM

0

X

T

Z
0

X

0

þ

cðh À hÞ

2

chðh À hÞ

Z

1

À


dx


ðhs À hs Þ dM þ dx

0

X

Z T Z
dt ðh À hÞ2 dx
6 ðaM À 2bL a À cL aM0þ þ cM bM0À Þ
X
0

Z T Z 
Z 1
þ cM b
dt
ðh À hÞ
ðhs À hs Þ dM À dx
X
0
0

Z T Z 
Z 1
þ
dt
ðh À hÞ
ðhs À hs Þ dM dx:
þ cM b
0

0

X

First, we consider the second term in the above:

Z 1
Z T Z 
À
dt
ðh À hÞ
ðhs À hs Þ dM dx
0

¼

Z
Z

X
1

dM À ðsÞ

0

Â

Z

1

À

dM ðsÞ
Z

Z

T

dt
0

1=2

2

ðhðt; xÞ À hðt; xÞÞ dx

X

1=2

2

ðhðt À s; xÞ À hðt À s; xÞÞ dx

dt
0

X

Z

Z

T

½ðhðt; xÞ À hðt; xÞÞðhðt À s; xÞ À hðt À s; xÞފ dx

dt
0

6
0

0
T

Z

:

X

Using the periodicity of hðt; xÞ and hðt; xÞ, we have
Z T Z
dt ðhðt À s; xÞ À hðt À s; xÞÞ2 dx
X
0
Z

Z T
Z
Z T Às
¼
dx
ðhðt À s; xÞ À hðt À s; xÞÞ2 dt ¼
dx
ðhðs; xÞ À hðs; xÞÞ2 ds
X
X
0
Às
Z
Z T
Z T Z
2
¼
dx
ðhðs; xÞ À hðs; xÞÞ ds ¼
dt ðhðt; xÞ À hðt; xÞÞ2 dx:
0

X

0

Then we obtain
Z T Z 
Z
dt

ðh À hÞ
0

X
1

0
T


ðhs À hs ÞdM À dx

Z
2
dM À ðsÞ
dt ðhðt; xÞ À hðt; xÞÞ dx
X
0
0
Z T Z
2
À
¼ M0
dt ðh À hÞ dx:
Z

Z

1

X

6

0

X


Y. Tang, L. Zhou / Appl. Math. Comput. 148 (2004) 453–460

459

Similarly, we can get the estimate of the third term:

Z T Z 
Z 1
þ
dt
ðh À hÞ
ðhs À hs ÞdM dx
0

X
1

Z

þ

Z

0
T

Z

2

dM ðsÞ
dt ðhðt; xÞ À hðt; xÞÞ dx
X
0
Z T Z
2
dt ðh À hÞ dx:
¼ M0þ

6

0

0

X

Therefore, we have
Z T Z
o
0¼

dt ðh À hÞ ðh À hÞ dx
ot
X
0

Z T Z
2
dt ðh À hÞ dx
6 ½aM À 2bL a À cL aM0þ þ cM bM0À þ cM bðM0þ þ M0À ފ
X
0
Z T Z
2
¼ ½aM À 2bL a À cL aM0þ þ cM bM0þ þ 2cM bM0À Š
dt ðh À hÞ dx 6 0:
0

X

ð2:12Þ
The last inequality follows from the assumption (2.4). Then (2.12) implies
that hðt; xÞ ¼ hðt; xÞ for ðt; xÞ 2 Rþ Â X. This means that hðt; xÞ ¼ hðt; xÞ ¼
uà ðt; xÞ is a T -periodic solution of (1.6) and (1.7). From (2.10), for any initial
function /ðt; xÞ satisfying a 6 / 6 b in RÀ Â X, the solution uðt; xÞ of (1.6)–(1.8)
satisfies
uðt; xÞ ! uà ðt; xÞ
as t ! 1ðx 2 XÞ. This completes the proof.

Ã

3. Discussion
The method of T -upper and lower solutions and the bootstrap technique of
periodic monotone iteration are very useful in the research of the periodic
systems. But in general, we just obtain the quasi-solutions of the periodic
boundary value problems, they are not the solutions of the problems. However,
for a pair of quasi-solutions hðt; xÞ and hðt; xÞ, the sector hhðt; xÞ; hðt; xÞi is an
attractor of the associated initial boundary value problem. Furthermore, if
hðt; xÞ ¼ hðt; xÞ, then the quasi-solution hðt; xÞ (or hðt; xÞ) is exactly the solution,
and the asymptotic behavior of the solutions to the associated initial boundary
value problem is also obtained. Therefore, the sufficient conditions of
hðt; xÞ ¼ hðt; xÞ can help us to get more detailed information about periodic
systems.


460

Y. Tang, L. Zhou / Appl. Math. Comput. 148 (2004) 453–460

In our main result Theorem 2.1, if we take l0 ðsÞ ¼ dðs À rÞ, aðt; xÞ ¼ 1 þ
eðt; xÞ, Àbe 6 e 6 be , bðt; xÞ  b, cðt; xÞ  c, where r, b, c are positive constants,
be > 0 is sufficiently small, then we have
M0þ ¼ 1;

M0À ¼ 0;

a¼

be
1
À

;
bþc bÀc

b¼

be
1
þ
bþc bÀc

and the conditions (2.2) and (2.4) become
b>c

1 þ be
;
1 À be

À

bÀc
bþc
be < 0:
þ be þ 2
bþc
bÀc

For sufficiently small be > 0, it is obvious that Theorem 1.3 is special case of
Theorem 2.1.

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