A sufficient condition for the existenceof periodic solution for a reaction Yanbin Tang *, Li Zhou Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 43
Trang 1A sufficient condition for the existence
of periodic solution for a reaction
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
buðt; xÞ
Z 1 0
kðsÞuðt s; xÞ ds
; ðt; xÞ 2 Rþ X;
ð1:1Þ
q
The project supported by National Natural Science Foundation of China.
*
Corresponding author.
E-mail address: tangyb@public.wh.hb.cn (Y Tang).
0096-3003/$ - see front matter Ó 2002 Elsevier Inc All rights reserved.
doi:10.1016/S0096-3003(02)00860-3
www.elsevier.com/locate/amc
Trang 2onðt; xÞ ¼ 0; ðt; xÞ 2 Rþ oX; ð1:2Þ uðt; xÞ ¼ /ðt; xÞ; ðt; xÞ 2 R X; ð1:3Þ where L ¼ ðo=otÞ M, a and b are positive constants, X is a bounded domain
in Rnwith its boundaryoX being a C2––manifold, /2 CðR XÞ is a bounded nonnegative function and /ð0; Þ 2 C1ðXÞ Redlinger obtained the following result:
Theorem 1.1 Suppose that k2 CðRþÞ \ L1ðRþÞ which k 6¼ 0 and b >
R1
0 jkðsÞj ds, then the initial boundary value problem (1.1)–(1.3) has a unique bounded, nonnegative and regular solution uðt; xÞ for each positive, continuous and bounded function / on R X Moreover, if /ð0; Þ 6¼ 0, then uðt; xÞ > 0 for allðt; xÞ 2 ð0; þ1Þ X and limt!1uðt; xÞ ¼ a=ðb þR1
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:
Luðt; xÞ ¼ uðt; xÞ aðt; xÞ
bðt; xÞuðt; xÞ cðt; xÞ
Z 1 0
uðt s; xÞ dlðsÞ
; ð1:4Þ where a, b and c are sufficiently smooth functions and 0 < a16aðt; xÞ 6 a2,
0 < b16bðt; xÞ 6 b2, 0 < c16cðt; xÞ 6 c2, lðÞ is of bounded variation with lð0Þ ¼ 0
Let MðtÞ denote the total variation of lðÞ on ½0; t and MðtÞ ¼
½MðtÞ lðtÞ=2 for all t 2 Rþ, then MðtÞ and MðtÞ are nonnegative and non-decreasing on Rþ Denote M0¼ limt!þ1MðtÞ, M
0 ¼ limt!þ1MðtÞ, l0¼ limt!þ1lðtÞ Shi and Chen obtained the following asymptotic behavior: Theorem 1.2 Suppose that b1> c1M
0, a1ðb1 c1M
0Þ > c2a2M0þand uðt; xÞ is a solution of (1.2)–(1.4) Then there exist positive constants a, bð0 < a 6 bÞ such that
a 6lim inf
t!þ1 min
x2X
uðt; xÞ 6 lim sup
t!þ1
max
x2X
uðt; xÞ 6 b:
In [3], Zhou and Fu considered a periodic logistic delay equation with dis-crete delay effect as following:
Luðt; xÞ ¼ uðt; xÞ½1 þ ðt; xÞ buðt; xÞ cuðt s; xÞ; ðt; xÞ 2 Rþ X;
ð1:5Þ where ðt; xÞ is T -periodic in t and there is a constant ^with 0 < ^ 1 such that
^ 6 ðt; xÞ 6 ^ on X ½0; T , b and c are positive constants They obtained the following conclusion
Trang 3Theorem 1.3 Let b > c and ^ be sufficiently small Then there is a unique T -periodic solution u in the sector J ¼ 1
bþc ^
bc; 1 bþcþ ^ bc
Moreover, for any initial function /ðt; xÞ 2 J , the corresponding solution uðt; xÞ of (1.5), (1.2), (1.3) satisfies uðt; xÞ ! uðt; xÞ as t ! þ1 uniformly in x 2 X
In this paper we consider the following periodic reaction diffusion equation with infinite delay:
Luðt; xÞ ¼ uðt; xÞ aðt; xÞ
bðt; xÞuðt; xÞ cðt; xÞ
Z 1 0
uðt s; xÞ dlðsÞ
; ð1:6Þ ou
onðt; xÞ ¼ 0; ðt; xÞ 2 Rþ oX; ð1:7Þ uðt; xÞ ¼ /ðt; xÞ; ðt; xÞ 2 R 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 g¼ gðt; xÞ which is continuous, strictly positive and
T-periodic in t on R X Let gM and gLdenote the maximum and minimum of
g on R 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 as-ymptotic 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 bLb cLaM0þþ cMbM0¼ 0;
aL bMa cMbM0þþ cLaM0¼ 0: ð2:1Þ
We assume that
aLbL> cMðaLM0þ aMM0þÞ: ð2:2Þ Then it is easy to solve (2.1) and a unique solution is given by
a¼ aLðbL cMM
0Þ aMcMMþ
0
ðbL cMM
0ÞðbM cLM
0Þ cLcMðMþ
0Þ2;
b¼ aMðbM cLM
0Þ aLcLM0þ
ðbL cMMÞðbM cLMÞ cLcMðMþÞ2:
ð2:3Þ
Trang 4Eq (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 2bLa cLaMþ
0 þ cMbMþ
0 þ 2cMbM
0 <0; ð2:4Þ then problem (1.6) and (1.7) has a unique T -periodic solution uðt; 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 ! uðt; xÞ; t! 1ðx 2 XÞ:
Proof We know that the problem (1.6)–(1.8) is equivalent to the following: Luðt; xÞ ¼ u a
bu c
Z 1 0
uðt s; xÞ dMþðsÞ þ c
Z 1 0
uðt s; xÞ dMðsÞ
; ð2:5Þ ou
onðt; xÞ ¼ 0; ðt; xÞ 2 Rþ oX; ð2:6Þ uðt; xÞ ¼ /ðt; xÞ; ðt; xÞ 2 R 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
such that
Lhðt; xÞ ¼ h a
bh c
Z 1 0
hðt s; xÞ dMþðsÞ þ c
Z 1 0
hðt s; xÞ dMðsÞ
; Lhðt; xÞ ¼ h a
bh c
Z 1 0
hðt s; xÞ dMþðsÞ þ c
Z 1 0
hðt s; xÞ dMðsÞ
; ðt; xÞ 2 Rþ X;
oh
onðt; xÞ ¼oh
onðt; xÞ ¼ 0; ðt; xÞ 2 Rþ oX:
ð2:9Þ Moreover, for any initial function /ðt; xÞ satisfying a 6 / 6 b in R X, the corresponding solution uðt; xÞ of (2.5)–(2.7) satisfies
Trang 5hðt; xÞ 6 uðt; xÞ 6 hðt; xÞ; t! 1ðx 2 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,
Lhðt; xÞ P h
a bh c
Z 1 0
hðt s; xÞ dlðsÞ
;
Lhðt; xÞ 6 h
a bh c
Z 1 0
hðt s; xÞ dlðsÞ
; ðt; xÞ 2 Rþ X;
oh
onðt; xÞ ¼oh
onðt; xÞ ¼ 0; ðt; xÞ 2 Rþ oX:
ð2:11Þ
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 T
0
dt
Z
X
ðh hÞo
otðh hÞ dx ¼
Z T 0
dt Z
X
ðh hÞ
Mh
þ h a
bh c
Z 1 0
hsdMþðsÞ þ c
Z 1 0
hsdMðsÞ
dx
Z T 0
dt
Z
X
ðh hÞ Mh
þ h a
bh c
Z 1 0
hsdMþðsÞ þ c
Z 1 0
hsdMðsÞ
dx
¼
Z T
0
dt Z
X
jrðh hÞj2dxþ
Z T 0
dt Z
X
aðh hÞ2dx
Z T
0
dt
Z
X
bðh þ hÞðh hÞ2dxþ
Z T 0
dt Z
X
cðh hÞ
h
Z 1 0
hsdMþþ h
Z 1 0
hsdMþ h
Z 1 0
hsdMþ h
Z 1 0
hsdM
dx
6ðaM 2bLaÞ
Z T 0
dt Z
X
ðh hÞ2dx
Z T
dt
Z cðh
hÞ2
Z 1
hsdMþ
dx
Trang 6Z T
0
dt
Z
X
cðh
hÞ2
Z 1 0
hsdM
dx þ
Z T
0
dt
Z
X
chðh
hÞ
Z 1 0
ðhs hsÞ dM
dx þ
Z T
0
dt
Z
X
chðh
hÞ
Z 1 0
ðhs hsÞ dMþ
dx
6ðaM 2bLa cLaMþ
0 þ cMbM
0Þ
Z T 0
dt Z
X
ðh hÞ2dx
þ cMb
Z T
0
dt Z
X
ðh
hÞ
Z 1 0
ðhs hsÞ dM
dx
þ cMb
Z T
0
dt Z
X
ðh
hÞ
Z 1 0
ðhs hsÞ dMþ
dx:
First, we consider the second term in the above:
Z T
0
dt
Z
X
ðh
hÞ
Z 1 0
ðhs hsÞ dM
dx
¼
Z 1
0
dMðsÞ
Z T 0
dt Z
X
½ðhðt; xÞ hðt; xÞÞðhðt s; xÞ hðt s; xÞÞ dx 6
Z 1
0
dMðsÞ
Z T 0
dt Z
X
ðhðt; xÞ
hðt; xÞÞ2dx
1=2
Z T
0
dt Z
X
ðhðt
s; xÞ hðt s; xÞÞ2dx
1=2
:
Using the periodicity of hðt; xÞ and hðt; xÞ, we have
Z T
0
dt
Z
X
ðhðt s; xÞ hðt s; xÞÞ2dx
¼
Z
X
dx
Z T 0
ðhðt s; xÞ hðt s; xÞÞ2dt¼
Z
X
dx
Z Ts
s
ðhðs; xÞ hðs; xÞÞ2ds
¼
Z
X
dx
Z T 0
ðhðs; xÞ hðs; xÞÞ2ds¼
Z T 0
dt Z
X
ðhðt; xÞ hðt; xÞÞ2dx:
Then we obtain
Z T
0
dt
Z
X
ðh
hÞ
Z 1 0
ðhs hsÞdM
dx 6
Z 1
0
dMðsÞ
Z T 0
dt Z
X
ðhðt; xÞ hðt; xÞÞ2dx
¼ M
0
Z T
dt Z
ðh hÞ2dx:
Trang 7Similarly, we can get the estimate of the third term:
Z T
0
dt
Z
X
ðh
hÞ
Z 1 0
ðhs hsÞdMþ
dx 6
Z 1
0
dMþðsÞ
Z T 0
dt Z
X
ðhðt; xÞ hðt; xÞÞ2dx
¼ Mþ
0
Z T
0
dt Z
X
ðh hÞ2dx:
Therefore, we have
0¼
Z T
0
dt
Z
X
ðh hÞo
otðh hÞ dx
6½aM 2bLa cLaMþ
0 þ cMbM
0 þ cMbðMþ
0 þ M
0Þ
Z T 0
dt Z
X
ðh hÞ2dx
¼ ½aM 2bLa cLaM0þþ cMbM0þþ 2cMbM0
Z T 0
dt Z
X
ðh hÞ2dx 6 0:
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 X This means that hðt; xÞ ¼ hðt; xÞ ¼
uðt; 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 R X, the solution uðt; xÞ of (1.6)–(1.8) satisfies
uðt; xÞ ! uðt; xÞ
as t! 1ðx 2 XÞ This completes the proof
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
Trang 8In our main result Theorem 2.1, if we take l0ðsÞ ¼ dðs rÞ, aðt; xÞ ¼ 1 þ eðt; xÞ, bee 6 e 6 bee, bðt; xÞ b, cðt; xÞ c, where r, b, c are positive constants, bee > 0 is sufficiently small, then we have
M0þ ¼ 1; M0 ¼ 0; a¼ 1
bþ c
bee
b c; b¼
1
bþ cþ
bee
b c and the conditions (2.2) and (2.4) become
b > c1þbee
1bee;
b c
bþ cþbee þ 2bbþ c cbee < 0:
For sufficiently smallbee > 0, it is obvious that Theorem 1.3 is special case of Theorem 2.1
References
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(1985) 135–142.
[2] B Shi, Y Chen, A prior bounds and stability of solutions for a Volterra reaction diffusion equation with infinite delay, Nonlinear Anal., TMA 44 (2001) 97–121.
[3] L Zhou, Y Fu, Existence and stability of periodic quasisolutions in nonlinear parabolic systems with discrete delays, J Math Anal Appl 250 (2000) 139–161.
[4] S Ahmad, A.C Lazer, Asymptotic behavior of solutions of periodic competition diffusion system, Nonlinear Anal., TMA 13 (1989) 263–284.
[5] C.V Pao, Quasisolutions and global attractor of reaction diffusion system, Nonlinear Anal., TMA 26 (1996) 1889–1903.
[6] C.V Pao, Periodic solutions of parabolic system with nonlinear boundary conditions, J Math Anal Appl 234 (1999) 695–716.