Asymptotic Behavior of a Periodic Diffusion System

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Journal of Inequalities and Applications Volume 2010, Article ID 764703,11pages doi:10.1155/2010/764703

Research Article

Asymptotic Behavior of a Periodic Diffusion System

Songsong Li

^{1, 2}

and Xiaofeng Hui

¹

1School of Management, Harbin Institute of Technology, Harbin 150001, China

2School of Finance and Economics Management, Harbin University, Harbin 150086, China

Correspondence should be addressed to Songsong Li,[email protected] Received 26 June 2010; Accepted 25 August 2010

Academic Editor: P. J. Y. Wong

Copyrightq2010 S. Li and X. Hui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the asymptotic behavior of the nonnegative solutions of a periodic reaction diﬀusion system. By obtaining a priori upper bound of the nonnegative periodic solutions of the corresponding periodic diﬀusion system, we establish the existence of the maximum periodic solution and the asymptotic boundedness of the nonnegative solutions of the initial boundary value problem.

1. Introduction

In this paper, we consider the following periodic reaction diﬀusion system:

∂u

∂t Δu^m¹b1u^α¹v^β¹, x, t∈Ω×R, 1.1

∂v

∂t Δu^m²b2u^α²v^β², x, t∈Ω×R, 1.2 with initial boundary conditions

ux, t vx, t 0, x, t∈∂Ω×R, 1.3

ux,0 u0x, vx,0 v0x, x∈Ω, 1.4

wherem1,m2 >1,α1,α2,β1,β2 ≥1,Ω⊂ Rⁿis a bounded domain with a smooth boundary

∂Ω,b1 b1x, tandb2 b2x, tare nonnegative continuous functions and ofT-periodic T >0with respect tot, andu0andv0are nonnegative bounded smooth functions.

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In dynamics of biological groups 1,2, the system1.1-1.2was used to describe the interaction of two biological groups without self-limiting, where the diffusion terms reflect that the speed of the diffusion is slow. In addition, the system 1.1-1.2 can also be used to describe diffusion processes of heat and burning in mixed media with nonlinear conductivity and volume release, where the functionsu, vcan be treated as temperatures of interacting components in the combustible mixture 3.

For case ofm1m21, we get the classical reaction diﬀusion system of Fujita type

∂u

∂t Δuu^α¹v^β¹, ∂v

∂t Δvu^α²v^β². 1.5

This type reaction diﬀusion system 1.5 models such as heat propagations in a two- component combustible mixture 4, chemical processes 5, and interaction of two biological groups without self-limiting 6,7. The problem about system1.5includes global existence and global existence numbers, blow-up, blow-up rates, blow-up sets, and uniqueness of weak solutionssee 8–10and references therein.

In this paper, we will work on the diﬀusion system1.1-1.2; for results about single equation, see 11–16 and so on. In the past two decades, the system 1.1-1.2 has been deeply investigated by many authors, and there have been much excellent works on the existence, uniqueness, regularity and some other qualitative properties of the weak solutions of the initial boundary value problemsee 17–22and references therein. Maddalena 20 especially, established the existence and uniqueness of the solutions of the initial boundary value problem 1.1–1.4, and Wang 22 established the existence of the nonnegative nontrivial periodic solutions of the periodic boundary value problem1.1–1.3whenmi>1, αi, βi≥1,and αi/m1 βi/m2<1,i1,2.

Our work is to consider the existence and attractivity of the maximal periodic solution of the problem1.1–1.3. It should be remarked that our work is not a simple work. The main reason is that the degeneracy of1.1,1.2makes the work of energy estimates more complicated. Since the equations have periodic sources, it is of no meaning to consider the steady state. So, we have to seek a new approach to describe the asymptotic behavior of the nonnegative solutions of the initial boundary value problem. Our idea is to consider all the nonnegative periodic solutions. We fist establish some important estimations on the nonnegative periodic solutions. Then by the De Giorgi iteration technique, we provide a priori estimate of the nonnegative periodic solutions from the upper bound according to the maximum norm. These estimates are crucial for the proof of the existence of the maximal periodic solution and the asymptotic boundedness of the nonnegative solutions of the initial boundary value problem.

This paper is organized as follows. In Section 2, we introduce some necessary preliminaries and the statement of our main results. InSection 3, we give the proof of our main results.

2. Preliminary

In this section, as preliminaries, we present the definition of weak solutions and some useful principles. Since 1.1 and1.2are degenerated wheneveru v 0, we focus our main eﬀorts on the discussion of weak solutions.

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Definition 2.1. A vector-valued function u, v is called to be a weak supsolution to the problem1.1–1.4inQτ Ω×0, τ withτ > 0 if|∇u^m¹|,|∇v^m²| ∈ L²Qτ, and for any nonnegative functionϕ∈C¹Q_τwithϕ|∂Ω× 0,τ0 one has

Ωux, τϕx, τdx−

Ωu0xϕx,0dx−

Qτ

u∂ϕ

∂tdxdt

Qτ

∇u^m¹∇ϕ dx dt≥

Qτ

b1u^α¹v^β¹ϕ dx dt,

Ωvx, τϕx, τdx−

Ωv0xϕx,0dx−

Qτ

v∂ϕ

∂tdx dt

Qτ

∇v^m²∇ϕ dx dt≥

Qτ

b2u^α²v^β²ϕ dx dt, ux, t≥0, vx, t≥0, x, t∈∂Ω×0, τ, ux,0≥u0x, vx,0≥v0x, x∈Ω.

2.1

Replacing “≥” by “≤” in the above inequalities follows the definition of a weak subsolution. Furthermore, if u, vis a weak supersolution as well as a weak subsolution, then we call it a weak solution of the problem1.1–1.4.

Definition 2.2. A vector-valued function u, v is said to be a T-periodic solution of the problem1.1–1.3if it is a solution such that

u·,0 u·, T, v·,0 v·, T a.e inΩ. 2.2

A vector-valued functionu, vis said to be aT-periodic supersolution of the problem1.1–

1.3if it is a supersolution such that

u·,0≥u·, T, v·,0≥v·, T a.e inΩ. 2.3

A vector-valued functionu, vis said to be aT-periodic subsolution of the problem1.1–

1.3if it is a subsolution such that

u·,0≤u·, T, v·,0≤v·, T a.e inΩ. 2.4

A pair of supersolutionu, vand subsolutionu, vare called to be ordered if

u≥u, v≥v a.e.inQ_T Ω×0, T. 2.5

Several properties of solutions of problem1.1–1.4are needed in this paper.

Lemma 2.3see 17. Ifαi ≥ 1, βi ≥ 1, αi/m1 βi/m2 < 1 with|Ω| < M0and M0 is a constant depending onmi, αi, βi, i = 1, 2, then there exist global weak solutions to1.1–1.4.

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Lemma 2.4see 20. Letting u, vbe a subsolution of the problem1.1–1.4with the initial value u₀, v₀, and letting u, vbe a supsolution of the problem1.1–1.4with the initial value u0, v0, thenu≤u,v≤va.e. inQTifu₀≤u0,v₀≤v0a.e. inΩ.

Lemma 2.5regularity 23. Letux, tbe a weak solution of

∂u

∂t Δu^mfx, t, m >1, 2.6

subject to the homogeneous Dirichlet condition 1.3. If f ∈ L^∞QT, then there exist positive constants K and β ∈ 0,1 depending only upon τ ∈ 0, T and f_∞ such that for any x1, t1,x2, t2∈Ω× τ, T, one has

|ux1, t1−ux2, t2| ≤K

|x1−x2|^β|t1−t2|^β/2

. 2.7

The main result of this paper is the following theorem.

Theorem 2.6. Ifmi >1, αi ≥ 1, βi ≥1, andαi/m1 βi/m2 <1 with|Ω|< M0 andM0 is a constant depending onmi, αi, βi,i1,2,then problem1.1–1.3has a maximal periodic solution U, Vwhich is positive inΩ. Moreover, ifu, vis the solution of the initial boundary value problem 1.1–1.4with nonnegative initial valueu0, v0, then for anyε >0, there existst1depending onu0

andε,t2depending onv0andε, such that

0≤u≤Uε, forx∈Ω, t≥t1,

0≤v≤Vε, forx∈Ω, t≥t2. 2.8

3. The Main Results

In this section, we first show some important estimates on the solutions of the periodic problem 1.1–1.3. Then, by the De Giorgi iteration technique, we establish the a prior upper bound of periodic solutions of1.1–1.3, which is used to show the existence of the maximal periodic solution of1.1–1.3and its attractivity with respective to the nonnegative solutions of the initial boundary value problem1.1–1.4.

Lemma 3.1. Letu, vbe nonnegative solution of1.1–1.3. Ifαi≥1, βi≥1,αi/m1βi/m2<

1 with|Ω| < M0 andM0 is a constant depending onmi, αi, βi,i 1,2,then there exists positive constantsrandslarge enough such that

α2

m2−β2 < m1r−1

m2s−1 < m1−α1

β1 , 3.1

u_LrQT≤C, v_LsQT≤C, 3.2

whereC >0 is a positive constant depending onm1,m2,α1,α2,β1,β2,r,s,and|Ω|.

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Proof. Forr >1, multiplying1.1byu^r−1and integrating overQT, by the periodic boundary value condition, we have

4r−1m1

m1r−1²

Ω

∇u^m¹^r−1/2²dx dt

QT

b1x, tu^α¹^r−1v^β¹dx dt, 3.3

that is,

Ω

∇u^m¹^r−1/2²dx dt≤ Cbm1r−1² 4r−1m1

QT

u^α¹^r−1v^β¹dx dt, 3.4

whereCbb1x, t

QT

. By the Poincar´e inequality, we have

Ωu^m_ε¹^r−1dx≤C

Ω

∇u^mε ¹^r−1/2²dx, 3.5

where Cis a constant depending only on |Ω| and N. Notice that α1/m1 β1/m2 < 1 impliesα1 < m1. Furthermore, we haveα1r−1< m1r−1. Then, by Young’s inequality, we obtain

u^α¹^r−1v^β¹ ≤ 1 2

r−1m1

CCb

2 m1r−1

2

u^m¹^r−1C1v^β¹^m¹^r−1/m¹^−α¹, 3.6

whereC1is the constant of Young’s inequality. Then, from3.4, we have

QT

u^m¹^r−1dx dt≤ 1 2

QT

u^m¹^r−1dx dtC1

QT

v^β¹^m¹^r−1/m¹^−α¹dx dt, 3.7

that is,

QT

u^m¹^r−1dx dt≤C1

QT

v^β¹^m¹^r−1/m¹^−α¹dx dt. 3.8

Similarly, we get an estimate forv^swiths >1, that is,

QT

v^m²^s−1dx dt≤C2

QT

u^α²^m²^s−1/m²^−β²dx dt. 3.9

Hence,

QT

u^m¹^r−1dx dt

QT

v^m²^s−1dx dt

≤C1

QT

v^β¹^m¹^r−1/m¹^−α¹dx dtC2

QT

u^α²^m²^s−1/m²^−β²dx dt.

3.10

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Notice that,αi/m1 βi/m2<1,i1,2, impliesα2β1<m1−α1m2−β2. Then there exist r≥max{2m1α1,2α2}ands≥max{2m2β2,2β1}such that

β1

m1−α1 < m2s−1

m1r−1 < m2−β2

α2 . 3.11

By Young’s inequality, we have

QT

u^α²^m²^s−1/m²^−β²dx dt≤ 1 2C2

QT

u^m¹^r−1dx dtC|QT|,

QT

v^β¹^m¹^r−1/m¹^−α¹dx dt≤ 1 2C1

QT

v^m²^s−1dx dtC|QT|.

3.12

Together with3.10, we obtain

QT

u^m¹^r⁻¹dx dt

QT

v^m²^s−1dx dt≤C. 3.13

Thus, we prove the inequality3.2.

Lemma 3.2. Letu, vbe nonnegative solution of1.1–1.3. Ifαi≥1, βi≥1,αi/m1βi/m2<

1 with|Ω|< M0andM0is a constant depending onmi, αi, βi,i 1,2,then one has

QT

|∇u^m¹|²dx dt≤C,

QT

|∇v^m²|²dx dt≤C, 3.14

whereC >0 is a positive constant depending onm1,m2,α1,α2,β1,β2,r,s,and|Ω|.

Proof. Multiplying1.1byu^m¹and integrating overQT, by H ¨older’s equality, we have

QT

|∇u^m¹|²dx dt≤

QT

u^α¹^m¹v^β¹dx dt

≤

QT

u^2α¹^m¹dx dt

1/2

QT

v^2β¹dx dt

1/2

.

3.15

Takingr≥max{2α1m1,2β2}, s≥max{2β2m2,2α1}, byLemma 3.1, we can obtain the first inequality in3.14. The same is true for the second inequality in3.14.

Before we show the uniform super bound of maximum modulus, we first introduce a lemma as followssee 24.

Lemma 3.3. Suppose that a sequenceyh, h0,1,2, . . .of nonnegative numbers satisfies the recursion relation

yh1≤cb^hy_h^1ε, h0,1, . . . , 3.16

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with some positive constantsc, εandb≥1. Then,

yh≤c^1ε^h^−1/εb^1ε^h^−1/ε²^−h/εy^1ε₀ ^h. 3.17

In particular, if

y0≤θc^−1/εb^−1/ε², b >1, 3.18

then,

yh≤θb^−h/ε, 3.19

and consequentlyyh → 0 forh → ∞.

Lemma 3.4. Letu, vbe a solution of 1.1–1.3. Ifαi≥1, βi ≥1,αi/m1 βi/m2<1 with

|Ω|< M0andM0is a constant depending onmi, αi, βi,i 1,2,then there is a positive constantC such that

u_L∞QT≤C, v_L∞QT≤C. 3.20

Proof. Letkbe a positive constant. Multiplying1.1byu−k^m¹and integrating overQT, we have

1 m11

QT

∂

∂tu−k^m¹¹dx dt

QT

∇u−k^m¹²dx dt

QT

b1x, tu^α¹v^β¹u−k^m¹dx dt,

3.21

wheres max{s,0}. Denote thatμk mes{x, t∈QT :ux, t> k}. ByLemma 3.1with randslarge enoughand H ¨older’s inequality, we have

1 m11

QT

∂

∂tu−k^m¹¹dx dt

QT

≤C

QT

u^α¹v^β¹_ξ dx dt

ξ

QT

u−k^m¹^ξdx dt

1/ξ

≤C

QT

u−k^m¹^ξηdx dt

1/ξη

μk^1−1/η1/ξ,

3.22

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where ξ, η > 1 are to be determined. Using the Nirenberg-Gagliardo inequality with Lemma 3.1, we have

QT

u−k^m¹^ξηdx dt

1/ξη

≤C

QT

θ/2

, 3.23

where

θ

1− 1 ξη

1 N− 1

21 ₋₁

∈0,1. 3.24

Substituting3.22and3.23in3.21, we have

QT

∇u−k^m¹²dxdt≤C

QT

θ/2

μk^1−1/η1/ξ. 3.25

Setting

wk

QT

∇u−k^m¹²dx dt, 3.26

from3.25we obtain

wk≤Cμk2/2−θ1−1/η1/ξ. 3.27

TakekhM2−2^−h,h0,1, . . ., andM >0 is to be determined. Then, we have

kh1−kh^m¹^ξημkh1≤

QT

u−kh^m¹^ξηdx dt≤Cwkh^ξηθ/2. 3.28

From3.26, we have

μkh1≤C2^hm¹^ξημkh^{θη−1/2−θ}Cb^hμkh^γ, 3.29

whereb2^m¹^ξηandγ η−1ξη−1N/2ξηN. For any constantξ > 1, takeηto be a positive constant satisfying

η >max

2,2ξN ξN −1

, 3.30

then we haveγ >1. ByLemma 3.1, we can selectMlarge enough such that

μk0 μM≤C^−1/γ−14^−1/γ−1². 3.31

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According toLemma 3.3, we haveμkh → 0, ash → ∞, which implies thatux, t≤2Min QT. The uniform estimate forvx, t_L∞QTmay be obtained by a similar method. The proof is completed.

Letμ, ψ be the first eigenvalue and its corresponding eigenfunction to the Laplacian operator−Δon some domainΩ⊃⊃Ωwith respect to homogeneous Dirichlet data. It is clear thatψ_x>0 for allx∈Ω.

Now we give the proof of the main results of this paper.

Proof ofTheorem 2.6. We first establish the existence of the maximal periodic solution Ux, t, Vx, tof the problem1.1–1.3. Define the Poincar´e mapping

T T1, T2:C Ω

×C Ω

−→C Ω

×C Ω

, Tu0x, v0x ux, T, vx, T,

3.32

whereux, t, vx, tis the solution of the initial boundary value problem1.1–1.4with initial valueu0x, v0x. A similar argument as that in 22shows that the mapT is well defined.

Letunx, t, vnx, tbe the solution of the problem1.1–1.4with initial value u0x, v0x ux, vx

K1ψ1, K2ψ2

, 3.33

whereK1, K2, ψ1,andψ2are taken as those in 22. Then, by comparison principle, we have unx, T, vnx, T Tⁿux, vx,

un1x, t≤unx, t≤ux, vn1x, t≤vnx, t≤vx. 3.34

A standard argument shows that there existu^∗x, v^∗x∈CΩ×CΩand a subsequence of{Tⁿux}, denoted by itself for simplicity, such that

u^∗x, v^∗x lim

n→ ∞Tⁿux, vx. 3.35 Similar to the proof of Theorem 4.1 in 25, we can prove thatUx, t, Vx, t, which is the even extension of the solution of the initial boundary value1.1–1.4with the initial value u^∗x, v^∗x, is a periodic solution of 1.1–1.3. For any nonnegative periodic solution ux, t, vx, tof1.1–1.3, byLemma 3.4, we have

ux, t≤C0, vx, t≤C0 forx, t∈QT. 3.36

Taking

K1≥ C0

min_x∈Ωϕ^1/m₁ ¹x, K2 ≥ C0

min_x∈Ωϕ^1/m₂ ²x, 3.37

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to be combined with the comparison principle andu^∗x ≥ux,0, v^∗x ≥vx,0, then we obtainUx, t≥ ux, t,Vx, t≥vx, t, which implies thatUx, t, Vx, tis the maximal periodic solution of1.1–1.3.

For any given nonnegative initial value u0x, v0x, let ux, t, vx, t be the solution of the initial boundary problem1.1–1.4, and letω1x, t, ω2x, tbe the solution of1.1–1.4with initial valueω1x,0, ω2x,0 R1ϕ1x, R2ϕ2x, whereR1, R2satisfy the same conditions asK1, K2and

R1≥ u0_L^∞

min_x∈Ωϕ^1/m₁ ¹x, R2≥ v0_L^∞

min_x∈Ωϕ^1/m₂ ²x. 3.38

For anyx, t∈QT,k0,1,2, . . ., we have

ux, tkT≤w1x, tkT, vx, tkT≤w2x, tkT. 3.39

A similar argument as that in 25shows that ω^∗₁x, t, ω^∗₂x, t

k→ ∞limω1x, tkT, lim

k→ ∞ω2x, tkT

, 3.40

andω^∗₁x, t, ω₂^∗x, tis a nontrivial nonnegative periodic solution of1.1–1.3. Therefore, for anyε >0, there existsk0such that

ux, tkT≤ω^∗₁x, t ε≤Ux, t ε,

vx, tkT≤ω^∗₂x, t ε≤Vx, t ε, 3.41

for anyk ≥k0andx, t∈Q_T. Taking the periodicity ofω^∗₁x, t,ω^∗₂x, t,Ux, t, andVx, t into account, the proof of the theorem is completed.

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