Hindawi Publishing Corporation
Boundary Value Problems
Volume 2009, Article ID 654539, 19 pages
doi:10.1155/2009/654539
Research Article
Global Behavior for a Diffusive Predator-Prey
Model with Stage Structure and Nonlinear Density
Restriction-II: The Case in R1
Rui Zhang,1, 2 Ling Guo,1 and Shengmao Fu1
1
2
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China
Correspondence should be addressed to Shengmao Fu, fusm@nwnu.edu.cn
Received 2 April 2009; Accepted 31 August 2009
Recommended by Wenming Zou
A Holling type III predator-prey model with self- and cross-population pressure is considered.
Using the energy estimate and Gagliardo-Nirenberg-type inequalities, the existence and uniform
boundedness of global solutions to the model are dicussed. In addition, global asymptotic stability
of the positive equilibrium point for the model is proved by Lyapunov function.
Copyright q 2009 Rui Zhang et al. 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.
1. Introduction
This paper is a continuation of Part I 1. In Section 3 of Part I, using the energy estimate and
bootstrap arguments, the global existence of solutions for a Holling type III cross-diffusion
predator-prey model with stage-structure has been discussed when the space dimension be
less than 6. However, to obtain the L∞ estimate for the population density w of predator
species, there is not cross-diffusion for w in Part I.
All diffusive predator-prey systems behave, more or less, in the same way, for both
semilinear and cross-diffusive models, at least for small values of the cross diffusivities.
Consequently, all the available information for linear diffusive models is essential to realize
the behavior of the most complicated cross-diffusive systems 2–17.
In this paper, we consider the following cross-diffusion system:
ut du α11 u2 α12 uv α13 uw
vt dv α21 uv α22 v2 α23 vw
xx
xx
βv − au − bu2 − cu3 −
u − v,
u2 w
,
1 u2
0 < x < 1, t > 0,
2
Boundary Value Problems
wt d3 w α31 uw α32 vw α33 w2
ux x, t vx x, t wx x, t 0,
ux, 0 u0 x,
vx, 0 v0 x,
xx
− kw − γw2 αu2 w
,
1 u2
x 0, 1, t > 0,
wx, 0 w0 x,
0 < x < 1,
1.1
where d, d3 , αij i, j 1, 2, 3, α, β, γ, a, b, c, and k are positive constants. Also, d, d3 are linear
diffusion coefficients of u, v, w, respectively, while αii i 1, 2, 3 are referred as self-diffusion
j, i, j 1, 2, 3 are cross-diffusion pressures. If α12 α21 α23 α31 pressures, and αij i /
α32 0, then 1.1 reduces to the system 1.4 of Part I.
Recently, the work in 18–20 studied the existence, uniform boundedness, and
uniform convergence of global solutions for the Lotka-Volterra cross-diffusion models
without stage-structure in the case that the space dimension n 1. In this paper, we
consider mainly the existence and uniform boundedness of global solutions for the model
1.1 with nonlinear density restriction and stage-structure. Moreover, global asymptotic
stability of the positive equilibrium point for 1.1 is proved by an important lemma
of 21. The proof is complete and complement the uniform convergence theorem in
18–20.
2. Global Existence and Uniform Boundedness
For simplicity, denote | · |k,p · Wpk 0,1 , | · |p · Lp 0,1 . The local existence result of
solutions to 1.1 is an immediate consequence of a series of papers 22, 23 by Amann.
Roughly speaking, if u0 , v0 , w0 ∈ Wp1 0, 1, p > 1, then 1.1 has a unique nonnegative
solution u, v, w ∈ C0, T , Wp1 0, 1 C∞ 0, T , C∞ 0, 1, where T ≤ ∞ is the maximal
existence time for the solution. If u, v, w satisfies
sup |u·, t|1,p , |v·, t|1,p , |w·, t|1,p : 0 < t < T < ∞,
2.1
then T ∞. If, in addition, u0 , v0 , w0 ∈ Wp2 0, 1, then u, v, w ∈ C0, ∞, Wp2 0, 1.
The main result in this section is as follows.
Theorem 2.1. Let u0 , v0 , w0 ∈ W22 0, 1, u, v, w is the unique nonnegative solution of 1.1 in its
maximal existence interval 0, T . Assume that
8α11 α21 α31 > α21 α213 α212 α31 ,
8α12 α22 α32 > α32 α221 α223 α12 ,
8α13 α23 α33 > α23 α231 α232 α13 .
2.2
Boundary Value Problems
3
Then there exists t0 > 0 and positive constants M, M which depend on d, d3 , αij i, j 1, 2, 3,
β, a, b, c, k, γ, α, such that
sup |u·, t|1,2 , |v·, t|1,2 , |w·, t|1,2 : t ∈ t0 , T ≤ M ,
2.3
max{ux, t, vx, t, wx, t : 0 ≤ x ≤ 1, t0 ≤ t < T } ≤ M,
2.4
and T ∞. In particular, if d, d3 ≥ 1, d3 /d ∈ d, d, where d ≤ 1 and d are positive constants,
then M , M depend on d, d, but do not depend on d, d3 ≥ 1.
The following Gagliardo-Nirenberg-type inequalities and corresponding corollary
play an importance role in the proof of Theorem 2.1.
Theorem 2.2 see 18. Let Ω ⊂ Rn be a bounded domain with ∂Ω ∈ Cm . For every function
u ∈ Wrm Ω, 1 ≤ q, r ≤ ∞, the derivative Dj u 0 ≤ j < m satisfies the inequality
j a
|u|q ,
D u ≤ C |Dm u|r |u|1−a
q
p
2.5
provided one of the following three conditions is satisfied: 1 r ≤ q, 2 0 < nr − q/mrq < 1, or
3 nr − q/mrq 1, and m − n/q is not a nonnegative integer, where 1/p j/n a1/r − m/n 1 − a/q, for all a ∈ j/m, 1, and the positive constant C depends on n, m, j, q, r, a.
Corollary 2.3. There exists a positive constant C such that
2/3
|u|2 ≤ C |ux |1/3
2 |u|1 |u|1 ,
1/2
|u|4 ≤ C |ux |1/2
2 |u|1 |u|1 ,
∀u ∈ W21 0, 1,
2.6
∀u ∈ W21 0, 1,
2.7
|u|1 ,
|u|11/21
|u|7/2 ≤ C |ux |10/21
2
1
2/5
|ux |2 ≤ C |uxx |3/5
2 |u|1 |u|1 ,
∀u ∈ W21 0, 1,
∀u ∈ W22 0, 1.
2.8
2.9
For simplicity, denote that C is Sobolev embedding constant or other kind of absolute constant.
Aj , Bj , Cj are some positive constants which depend on αij i, j 1, 2, 3, β, a, b, c, k, γ, α. Also, Kj
are positive constants which depend on αij i, j 1, 2, 3, β, a, b, c, k, γ, α, d, d3 . When d, d3 ≥ 1, Kj
do not depend on d, d3 , but on d, d.
Proof of Theorem 2.1
Step 1. Estimate |u|1 , |v|1 ,|w|1 . Firstly, taking integration of the first and second equations
in 2.7 over the domain 0, 1, respectively, and combining the two integration equalities
4
Boundary Value Problems
linearly, we have
d
dt
1
1
1
βu − bu2 dx.
u a β v dx ≤ −a vdx 0
0
2.10
0
From Young inequality and Hölder inequality, we can see
d
dt
1
u a β v dx ≤ C1 −
0
a
aβ
1
u a β v dx,
2.11
0
where C1 1/4bβ a/a β2 . From which it follows that there exists a constant τ0 > 0,
such that
1
1
udx,
0
vdx ≤ M0 ,
t ≥ τ0 ,
2.12
0
where M0 2C1 a β/a max{a β−1 , 1}.
Secondly, taking integration of the third equations in 2.7 over domain 0, 1, we have
d
dt
1
wdx ≤ α − k
1
0
wdx − γ
1
0
2
wdx
.
2.13
0
This implies that there exists a constant τ0 > 0, such that
1
2|α − k|
,
γ
wdx ≤
0
t ≥ τ0 .
2.14
Let M1 max{M0 , 2|α − k|/γ}, τ1 max{τ0 , τ0 }. Then
1
1
udx,
0
1
vdx,
0
wdx ≤ M1 ,
t ≥ τ1 .
2.15
0
Moreover, there exists a positive constant M1 which depends on β, a, b, c, k, γ, α and the L1 norm of u0 , v0 , w0 , such that
1
1
udx,
0
1
vdx,
0
0
wdx ≤ M1 ,
t ≥ 0.
2.15 Boundary Value Problems
5
Step 2. estimate |u|2 , |v|2 and |w|2 . Multiplying the first three inequalities of Corollary 2.3 by
u, v, w, respectively, and integrating over 0, 1, we have
1 d
2 dt
1
u2 dx
0
≤ −d
1 d
2 dt
1
1
1
2α11 u α12 v α13 wu2x α12 uux vx α13 uux wx dx β uvdx,
0
0
v2 dx
≤ −d
1 d
2 dt
u2x dx −
0
0
1
1
1
vx2 dx
0
1
1
2
−
α21 u 2α22 v α23 wvx α21 vux vx α23 vvx wx dx uvdx,
0
0
w2 dx
0
≤ −d3
1
0
wx2 dx
−
1
0
α31 u α32 v 2α33 wwx2 α31 wux wx α32 wvx wx dx.
2.16
Let d min{d, d3 }. By the above three inequalities and Young inequality, we have
1 d
2 dt
1
u2 v2 w2 dx
0
≤ −d
1
0
u2x
vx2
wx2
dx −
1
0
qux , vx , wx dx 1 β1
α
u2 v2 w2 dx,
2
0
2.17
where
qux , vx , wx 2α11 u α12 v α13 wu2x α21 u 2α22 v α23 wvx2 α31 u α32 v 2α33 wwx2
α12 u α21 vux vx α13 u α31 wux wx α23 v α32 wvx wx
2.18
is quadratic form of ux , vx , wx . It is not hard to verify that qux , vx , wx is positive definite if
2.2 holds. Moreover, if 2.2 holds, then
1 d
2 dt
1
0
1
1 β1
u2 v2 w2 d ≤ −d
u2x vx2 wx2 dx u2 v2 w2 dx.
α
2
0
0
2.19
Now we proceed in the following two cases.
1
1
i It holds that t ≥ τ1 . By 2.6 and 2.15, we have 0 u2x dx ≥ 1/CM14 0 u2 dx3 − M12 ,
6
Boundary Value Problems
and
−d
1
0
u2x
vx2
wx2
dx ≤ 3d
M12
− C2 d
1
u v w
2
2
2
3
2.20
dx .
0
By 2.19 and 2.20, we can see that
1 d
2 dt
1
u2 v2 w2 dx
0
≤ −C2 d
1
u v w
2
2
2
3
dx
0
2.21
1 β1
α
u2 v2 w2 dx 3d M12 .
2
0
Thus, there exists positive constants τ2 > τ1 and M2 depending on d, d3 , β, a, b, c, k, γ, α, such
that
1
u2 dx,
1
0
v2 dx,
1
0
0
w2 dx ≤ M2 ,
t ≥ τ2 .
2.22
Since the zero point of the right-hand side in 2.21 can be estimated by positive constants
independent of d , when d ≥ 1. Thus M2 do not depend on d ≥ 1.
ii t ≥ 0. Repeating estimates in i by 2.9 , we can obtain that there exists a positive
constant M2 depending on d, d3 , β, a, b, c, k, γ, α and the L1 , L2 -norm of u0 , v0 , w0 , such that
1
u2 dx,
0
1
v2 dx,
1
0
0
w2 dx ≤ M2 ,
t ≥ 0,
2.22 t d1 t,
2.23
when d ≥ 1, M1 is independent of d .
Step 3. Estimate |ux |2 , |vx |2 ,|wx |2 . Introduce the scaling that
u
u
,
d1
v v
,
d1
w
w
,
d1
, v , w,
t by u, v, w, t, respectively. Then 2.7 reduces to
denote η d3 /d, and redenote u
ut Pxx fu, v, w,
0 < x < 1, t > 0,
vt Qxx gu, v, w,
0 < x < 1, t > 0,
wt Rxx hu, v, w,
0 < x < 1, t > 0,
ux x, t vx x, t wx x, t 0,
ux, 0 u
0 x,
vx, 0 v0 x,
2.24
x 0, 1, t > 0,
wx, 0 w
0 x,
0 < x < 1,
Boundary Value Problems
7
where P u α11 u2 α12 uv α13 uw, Q v α21 uv α22 v2 α23 vw, R ηw α31 uw α32 vw α33 w2 , fu, v, w βd−1 v − ad−1 u − bu2 − cdu3 − du2 w/1 d2 u2 , gu, v, w d−1 u − v,
hu, v, w −kd−1 w − rw2 αdu2 w/1 d2 u2 . We still proceed in following two cases.
i It holds that t ≥ τ2∗ dτ2 . From 2.15 and 2.22, we can easily obtain that
1
1
udx,
1
0
1
vdx,
0
2
1
u dx,
0
wdx ≤ M1 d−1 ,
0
1
2
v dx,
0
2.25
w2 dx ≤ M2 d−2 ,
0
|P |1 , |Q|1 , |R|1 ≤ DK1 d−1 ,
where K1 2 η M2 d−2 , D max{M1 , α11 α12 α13 , α21 α22 α23 , α31 α32 α33 }.
Multiply the first three equations in 2.24 by Pt , Qt , Rt and integrate them over 0, 1,
respectively, then adding up the three new equations, we have
1 y t ≤ −
2
1
0
u2t dx −
1
1
0
vt2 dx − η
1
0
wt2 dx −
1
qut , vt , wt dx
0
1 2α11 u α12 v α13 wut f α12 uvt f α13 uwt f dx
0
1
α21 vut g 1 α21 u 2α22 v α23 wvt g α23 vwt g dx
2.26
0
1
α31 wut h α32 wvt h η α31 u α32 v 2α33 w wt h dx,
0
1
where y 0 Px2 Qx2 R2x dx. It is not hard to verify by 2.4 that there exists a positive
constant C3 depending only on αij i, j 1, 2, 3, such that
qut , vt , wt ≥ C3 u v w u2t vt2 wt2 .
2.27
Thus,
1
1
1
1
1 2
2
2
y t ≤ − ut dx − vt dx − η wt dx − C3 u v w u2t vt2 wt2 dx
2
0
0
0
0
1
1
1 2α11 u α12 v α13 wut fdx 1 α21 u 2α22 v α23 wvt gdx
0
1
0
η α31 u α32 v 2α33 w wt hdx 0
1
0
α21 vut gdx 1
0
α23 vwt gdx 1
0
1
α12 uvt fdx 0
α31 wut hdx 1
2.28
1
α13 uwt fdx
0
α32 wvt hdx.
0
8
Boundary Value Problems
Using Young inequality, Hölder inequality and 2.24, we can obtain the following estimates:
1
u dx ≤
3
1
u dx ≤
4
1
1
u dx ≤
u dx ≤
6
uvdx ≤
u vdx ≤
2
u vdx ≤
u6 vdx ≤
1
7
1
4
uut dx ≤
1/5 1
0
7
1
u ut dx ≤
2
0
3
u4 ut dx ≤
0
1
2
1
1
uvut dx ≤
2
0
u2 vut dx ≤
1
2
,
2/5
7
u dx
,
0
1
3/5
7
u dx
,
0
1
4/5
7
u dx
,
0
1
1
u vdx 2
0
udx 0
1
2
3
u dx 2
0
1
5
u7 dx 0
1
1
2
0
0
1
0
1
0
u vdx 2
0
1
0
2
u4 vdx 2
≤
v dx
6
7
6
M24/5 d−8/5
1/2
0
uu2t dx ≤
1
≤
2
M23/5 d−6/5
1
1/5
7
u dx
,
0
1
2/5
7
u dx
u7 v7 dx,
0
1
1
M1 d−1 2
2
1/5
7
u dx
0
1
uu2t dx
1
M4/5 d−8/5
≤
2 2
uu2t dx
1
M2/5 d−4/5
≤
2 2
0
3
7
1
u7 v7 dx,
0
uu2t dx,
1
1/5
7
u dx
0
1
3/5
7
u dx
0
2
2
1
0
1
0
uu2t dx,
uu2t dx,
uu2t dx,
1
0
1
0
vu2t dx
vu2t dx
,
0
1
1
u vdx ≤ M24/5 d−8/5
2
0
1
u dx 2
0
1
v7 dx ≤
0
2
1
1/10 1
2
u dx
1
7
1/2
2
v dx
0
u7 dx 1
1
u ut dx ≤
2
0
1
0
2/5 1
0
2
3/10 1
2
u dx
u dx
1
2
u dx
0
≤ M2 d−2 ,
0
1
1
7
1/2
2
v dx
u dx
6
7
≤
M21/5 d−2/5
1/5
1
0
1
1
u vdx ≤
2
0
0
1/2 1
0
0
1
2
M22/5 d−4/5
1/5
2
u dx
0
3
≤
0
u dx
0
1
4/5 1
0
0
1
7
M23/5 d−6/5
2/5
2
u dx
u dx
0
≤
0
1
1
1
3/5 1
u dx
1
1
7
M24/5 d−8/5
3/5
2
u dx
0
0
1
2/5 1
0
1
0
1
7
u dx
0
≤
0
0
5
4/5
2
u dx
0
0
1
1/5 1
u dx
0
1
7
1
M4/5 d−8/5
≤
2 2
1
1/5
7
u dx
0
2
1
0
vu2t dx,
Boundary Value Problems
9
1
M4/5 d−8/5
≤
4 2
1
1
1
2
u3 vut dx ≤
0
1
1/5
7
u dx
0
2
u6 vdx 0
1
0
vu2t dx ≤
1
3
14
3
14
1 2
u7 v7 dx vu dx,
2 0 t
0
1
0
1 2
u7 v7 dx vu dx.
2 0 t
2.29
Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms
on the right-hand side of 2.28, we have
−
1
0
−
1
0
−η
1
0
u2t dx
1
≤−
2
vt2 dx ≤ −
wt2 dx ≤ −
1
1
2
1
0
1
η
2
0
0
2 −2
1
1
0
h2 dx,
0
2 −2
v dx a d
2abd−1
2ad
1
2
0
−2
g 2 dx,
0
R2xx dx η
1
f 2 dx,
0
2
Qxx
dx 1
f dx ≤ β d
2
2
Pxx
dx
1
1
1
u dx b
2
2
1
0
u3 dx 2ac
1
0
0
u4 dx d−2
0
uwdx 2bd
−1
1
0
u dx 2bcd
4
1
2
1
u dx c d
5
2 4
0
w2 dx
0
1
u wdx 2 u3 wdx
2
0
0
2
0
1
2/5
3/5 −6/5
7
2ac b 2 M2 d
u dx
2
0
1
2bcM22/5 d1/5
g 2 dx ≤ d−2
η
3/5
c
7
u dx
−2
h dx ≤ d η k α
2
2
0
1
0
−1
w dx 2kd γη
2
0
≤ η k α M2 d
γ
2
M21/5 d8/5
1
4/5
7
u dx
0
u2 v2 dx ≤ 2M2 d−4 ,
0
2
1
1
0
1
u6 dx
0
1
1/5
3/5 −13/5
−4
7
≤ a β 1 2a M2 d 2ba 1M2 d
u dx
1
2
2
2
ηM23/5 d−6/5
−4
1
w dx γ η
3
2
0
2kγηM24/5 d1−13/5
1
2/5
7
w dx
0
.
1
w4 dx
0
1
1/5
7
w dx
0
,
2.30
10
Boundary Value Problems
Thus
−
1
0
u2t dx
1
≤−
2
1
0
−
0
vt2 dx
2
Pxx
dx
C5 d
1
−1
−η
1
0
1
−
2
1
0
wt2 dx
η
−
2
2
Qxx
dx
1 η M24/5 d−8/5
C7 M22/5 d1/5
1
1
1
0
R2xx dx C4 2 η M2 d−4
1/5
u w dx
7
C6 1 η
7
M23/5 d−6/5
0
3/5
7
u dx
0
C8 M21/5 d8/5
1
1
2/5
u w dx
5
7
0
4/5
7
u dx
.
0
2.31
For the other terms on the right-hand side of 2.28, we have
1
1
1
−1 2
ut fdx ≤ βd ut vdx ad uut dx b u ut dx
0
0
0
0
1
−1 1
1
3
−1 cd u ut dx d wut dx
0
0
β2 a2 1
b2
≤
M1 d−3 M24/5 d−8/5
2
2
c2
M22/5 d6/5
2
3/5
7
u dx
0
3
2
1/5
7
u dx
0
1
0
uu2t dx
βd−1
2
1
0
vu2t dx
1
2
1
0
wu2t dx,
1
1
−1 2
uut fdx ≤ 2α11 βd uut vdx 2α11 ad u ut dx
0
0
0
1
2α11
1
1
−1 1
2α211 b
1
1
4
−1 u ut dx 2α11 dc u ut dx 2α11 d uut wdx
0
0
0
3
1
1
1/5
3/5
3
α211 β2 a2 1
α211 b2
4/5 −18/5
2/5 −4/5
7
7
≤
M2 d
u dx
M2 d
u dx
0
0
α2 c2
11 d2
1
0
u dx 3
7
1
0
uu2t dx
1
0
vu2t dx
1
0
wu2t dx,
Boundary Value Problems
1
1
1
1
α12 vut fdx ≤ α12 βd−1 ut v2 dx α12 ad−1 uvut dx α212 b u2 vut dx
0
0
0
0
1
1
3
−1 α12 dc u vut dx α12 d vwut dx
0
0
α2
≤ 12
2
2 −2
a d
b2
2
M24/5 d−8/5
1
11
1/5
7
u dx
0
1
1/5
α212 β2 1
4/5 −18/5
7
M2 d
v dx
2
0
1
3α212 b2
5 1
2 2
c d
u7 v7 dx vu2t dx,
7
2
2 0
0
1
1
1
1
α13 wut fdx ≤ α13 βd−1 vwut dx α13 ad−1 uwut dx α213 b u2 wut dx
0
0
0
0
1
1
3
−1 2
α13 dc u wut dx α13 d w ut dx
0
0
1
1/5
α213 a2 d−2 b2 /2
4/5 −8/5
7
≤
M2 d
u dx
2
0
⎡ 1/5 1
1/5 ⎤
1
α213 4/5 −18/5
⎣β
⎦
M d
v7 dx
w7 dx
2 2
0
0
1
3α213 b2
5 1
2 2
c d
u7 w7 dx wu2t dx,
7
2
2 0
0
1
M d−3 1 1
1
1
uvt2 vvt2 dx,
vt gdx ≤ d−1 uvt dx d−1 vvt dx ≤
0
0
2 0
0
1
1
1
−1 2
−1 α21 uvt gdx ≤ α21 d u vt dx α21 d uvvt dx
0
0
0
1
1/5
α221 4/5 −18/5
1 2
7
≤
M2 d
uvt vvt2 dx,
u dx
2 0
0
1
1
1
−1 −1 2
2α22 vvt gdx ≤ 2α22 d uvvt dx 2α22 d v vt dx
0
0
0
≤
⎡
1/5 1
1/5 ⎤
1
1
⎦ vvt2 dx,
M4/5 d−18/5 ⎣
u7 dx
v7 dx
2
0
1
1
−1 wvt gdx ≤ α23 d uwvt dx α23 d vwvt dx
0
0
0
1
α23
1
α22 0 vvt2 dx
−1 0
0
12
Boundary Value Problems
⎡
1
1/5 1
1/5 ⎤
1
1
α23 0 vvt2 dx 4/5 −18/5
7
7
⎣
⎦ wvt2 dx,
M2 d
≤
u dx
v dx
2
0
0
0
1
1
2
η wt hdx ≤ d ηα k wwt dx γη w wt dx
0
0
0
1
−1
γ2
α k2 2 −3
≤
η d M1 η2 M24/5 d−8/5
2
2
1/5
7
w dx
η
1
0
0
wwt2 dx,
1
1
2
uwt hdx ≤ α kd α31 uwwt dx α31 γ uw wt dx
0
0
0
1
α31
1
−1
⎡
1/5 1
1/5 ⎤
1
2
α231
γ
⎦
≤
M24/5 d−8/5 ⎣
u7 dx
w7 dx
α k2 d−2 2
2
0
0
1 1 2
1 1
u w dx uwt dx wwt2 dx,
2 0
2 0
0
1
1
1
−1
2
α32 vwt hdx ≤ α kd α32 vwwt dx α32 γ vw wt dx
0
0
0
3α2 γ 2
31
14
1
7
7
⎡
1/5 1
1/5 ⎤
1
2
α232
γ
⎦
≤
M24/5 d−8/5 ⎣
v7 dx
w7 dx
α k2 d−2 2
2
0
0
1 1
1 1
2
v w dx vwt dx wwt2 dx,
2 0
2 0
0
1
1
1
−1
2
3
2α33 wwt hdx ≤ 2α kd α33 w wt dx 2α33 γ w wt dx
0
0
0
3α2
32 γ 2
14
1
7
7
α k2 α233 4/5 −18/5
≤
M2 d
1
0
1/5
7
w dx
0
α2 γ 2
33 M22/5 d−4/5
1
3/5
7
w dx
0
wwt2 dx,
1
1
1
−1 2
3
uvt fdx ≤ α12 βd uvvt dx α12 ad u vt dx α12 b u vt dx
0
0
0
0
1
α12
1
−1 1
1
4
−1 α12 cd u vt dx α12 d uwvt dx
0
0
1
1
1/5
3/5
α212 β2 a2 1
α212 b2 2/5 −4/5
4/5 −18/5
7
7
≤
M2 d
M2 d
u dx
u dx
2
2
0
0
Boundary Value Problems
13
1
1
1
α212 c2 2 1 7
2
2
2
d
3 uvt dx vvt dx wvt dx ,
u dx 2
2
0
0
0
0
1
1
1
1
−1 −1 2
3
α13 uwt fdx ≤ α13 βd uvwt dx α13 ad u wt dx α13 b u wt dx
0
0
0
0
1
α313 cd
1
−1 u wt dx α13 d uwwt dx
0
0
4
1
1
1/5
3/5
α213 β2 a2 1
α213 b2 2/5 −4/5
4/5 −18/5
7
7
≤
M2 d
M2 d
u dx
u dx
2
2
0
0
α213 c2 2 1 7
3 1
1 1
1 1
d
u dx uwt2 dx vwt2 dx wwt2 dx,
2
2 0
2 0
2 0
0
1
1
1
−1 −1 2
α21 vut gdx ≤ α21 d uvut dx α21 d v ut dx
0
0
0
⎡
1/5 1
1/5 ⎤
1
1
α221 4/5 −18/5
7
7
⎣
⎦
≤
vu2t dx,
M d
u dx
v dx
2 2
0
0
0
1
1
−1 2
vwt gdx ≤ α23 d uvwt dx α23 d v wt dx
0
0
0
1
α23
−1 ⎡
1/5 1
1/5 ⎤
1
1
α223 4/5 −18/5
7
7
⎣
⎦
≤
vwt2 dx,
M d
u dx
v dx
2 2
0
0
0
1
1
wut hdx ≤ α31 α kd−1 w2 ut dx α31 γ w3 ut dx
0
0
0
1
α31
α k2 α231 4/5 −18/5
≤
M2 d
2
α2
31 γ 2 M22/5 d−4/5
2
1
1/5
7
w dx
0
3/5
7
w dx
0
1
0
wu2t dx,
1
1
2
3
wvt hdx ≤ α32 α kd w vt dx α32 γ w vt dx
0
0
0
1
α32
1
−1 1
1/5
2
α k2 α232 4/5 −18/5
7
≤
M2 d
w dx
2
0
α2 γ 2
32 M22/5 d−4/5
2
1
3/5
7
w dx
0
1
0
wvt2 dx.
2.32
14
Boundary Value Problems
Thus
1
1 2α11 α12 v α13 ut fdx 0
1
1 α21 u 2α22 v α23 wvt gdx
0
η α31 u α32 v 2α33 w wt hdx 0
1
1
α21 vut gdx 1
0
α23 vwt gdx 1
0
1
1
α12 uvt fdx 0
α31 wut hdx 0
1
α13 uwt fdx
0
1
α32 wvt hdx
0
C9
M1 d−3 2 η2
u v w u2t vt2 wt2 dx 0
1/5
1
C10 4/5 −8/5 −2
2
7
7
7
2d η
M2 d
u v w dx
0
≤ λ
1
C11 4/5 −8/5 M2 d
1 d2
2.33
3/5
u v w dx
7
7
7
0
1
C12 1 d2
u7 v7 w7 dx,
0
where λ is a positive constant.
Note by 2.8 and 2.9 that |P |7/2
7/2
B1 K14/3 d−4/3 |Pxx |22
−
1
2
1
0
2
Pxx
dx −
1
2
1
0
K12 d−2 ,
2
Qxx
dx −
≤
11/6
10/3
C|Px |5/3
|P |7/2
2 |P |1
1 , |Px |2
≤
and
η
2
1
0
R2xx dx ≤ −B2 min 1, η K1−4/3 d4/3 y 5/3 K12 d−2 2 η . 2.34
Choose a small enough number > 0, such that λ < C3 . According to 2.28–2.34, we have
1 y t ≤ −A1 min 1, η K1−4/3 y5/3 A2 K2 y1/6 A3 K3 y1/3 A4 K4 y1/2
2
A5 K5 y
2/3
A6 K6 y
5/6
2.35
A7 K7 ,
1
where y 0 dPx 2 dQx 2 dRx 2 dx.
3 depending
However, 2.35 implies that there exist positive constants τ3 > 0 and M
on d, d3 , αij i, j 1, 2, 3, β, a, b, c, k, γ, α, such that
1
0
2
dPx dx,
1
0
2
dQx dx,
1
0
3 ,
dRx 2 dx ≤ M
t ≥ τ3 .
2.36
Boundary Value Problems
15
When d, d3 ≥ 1, η ∈ d, d, the coefficients of 2.35 can be estimated by constants depending
3 depends on αij i, j on d, d, but not on d, d3 . Thus, when d, d3 ≥ 1, η ∈ d, d, M
1, 2, 3, β, a, b, c, k, γ, α,d, d, and is irrelevant to d, d3 ≥ 1. Since
⎛
Px
⎞
⎛
P u Pv Pw
⎞⎛
ux
⎞
⎟⎜ ⎟
⎜ ⎟ ⎜
⎜Qx ⎟ ⎜Qu Qv Qw ⎟⎜ vx ⎟,
⎠⎝ ⎠
⎝ ⎠ ⎝
Rx
Ru Rv Rw
wx
2.37
similar to 2.26 in 24, we have
|dux | |dvx | |dwx | ≤ D|dPx | |dQx | |dRx |,
0 < x < 1, t > 0,
2.38
where D is a positive constant only depending on η, αij i, j 1, 2, 3. Scaling back with 2.22
to original variable u, v, w, t and combining 2.36,2.38, there exist positive constants τ3 > 0
and M3 depending on d, d3 , αij i, j 1, 2, 3, β, a, b, c, k, γ, α, such that
1
0
u2x dx,
1
0
vx2 dx,
1
0
wx2 dx ≤ M3 ,
t ≥ τ3 .
2.39
In addition, when d, d3 ≥ 1, η ∈ d, d, M3 is dependent of d, d, but independent of d, d3 ≥ 1.
ii It holds that t ≥ 0. Replacing M1 , M2 with M1 , M2 in 2.24–2.34, we can obtain
that there exists a positive constant M3 depending on d, d3 , αij i, j 1, 2, 3, β, a, b, c, k, γ, α
and the W21 -norm of u0 , v0 , w0 such that
1
0
u2x dx,
1
0
vx2 dx,
1
0
wx2 dx ≤ M3 ,
t ≥ 0.
2.39 When d, d3 ≥ 1, η ∈ d, d, M3 is dependent of d, d, but independent of d, d3 ≥ 1.
Concluding from 2.15, 2.22, 2.39, and Sobolev embedding theorem, there exists
a positive constants t0 > 0, M, M depending on d, d3 , αij i, j 1, 2, 3, β, a, b, c, k, γ, α, such
that 2.3 and 2.4 are satisfied. Furthermore, when d, d3 ≥ 1, η ∈ d, d and the time t is
large enough, M, M are dependent of αij i, j 1, 2, 3, β, a, b, c, k, γ, α, d, d, but independent
of d, d3 ≥ 1.
Similarly, according to 2.15 , 2.22 , 2.39 , we can see that there exists a positive
constant M depending on d, d3 , αij i, j 1, 2, 3, β, a, b, c, k, γ, α and the initial functions
u0 , v0 , w0 , such that
|u·, t|1,2 , |v·, t|1,2 , |w·, t|1,2 ≤ M ,
t ≥ 0.
2.40
16
Boundary Value Problems
When d, d3 ≥ 1, η ∈ d, d, M is dependent of d, d, but independent of d, d3 . Thus T ∞.
This completes proof of Theorem 2.1.
3. Global Stability
From 1, we know that if
$
k
< m0 ,
α−k
√
24 β − a c2
β − a − c b 2 b p1
≤
2
√ ,
2
8c
24c
3b 4c β − a − c − b p1
α > k,
β > a,
H
where p1 9b2 24cβ − a − c ≥ 0, then 1.1 has the unique position equilibrium point
E∗ u∗ , v∗ , w∗ .
Theorem 3.1. Assume that all conditions in Theorem 2.1 and H are satisfied. Assume further that
1
β
u∗2 a bu∗ cu∗2 > 2 √
1 u∗2
8
2
u∗4
,
2
γ
1
>
,
α 21 u∗2 2
3.1
2
4 ∗ 2
1
1
w d d3 >
α23 M2 α32 w∗ d 2α11 M α12 M α13 M
αβ
β
α
1
1
α13 M2 α31 w∗
β
α
2
d α21 M 2α22 M α23 M
3.2
2
1 1
α12 α21 M2 w∗ d3 α31 M α32 M 2α33 M
α β
hold, where M is the positive constant in 2.4. Then the unique positive equilibrium point E∗ of 1.1
is globally asymptotically stable.
Remark 3.2. Since M is independent of d, d3 in the case of d, d3 ≥ 1, 3.2 is always satisfied if
d and d3 are big enough.
Proof. Define the Lyapunov function
1
Hu, v, w 2β
1
1
u − u dx 2
0
∗ 2
1
1
v − v dx α
0
∗ 2
1
w − w∗ − w∗ ln
0
w
dx.
w∗
3.3
Boundary Value Problems
17
≡ 0.
Let u, v, w be any solution of 1.1 with initial functions u0 x, v0 x, w0 x ≥ /
From the strong maximum principle for parabolic equations, it is not hard to verify that
u, v, w > 0 for t > 0. Thus
dH
≤−
dt
1%
1
d 2α11 u α12 v α13 wu2x d α21 u 2α22 v α23 wvx2
β
0
w∗ 2
1
1
α12 u α21 v ux vx
d3 α31 u α32 v 2α33 w 2 wx α
β
w
&
w∗
w∗
1
1
1
α13 u α31
ux wx α23 v α32
vx wx dx
β
α
w
α
w
1'
%
&
wu u∗ 1
−
u − u∗ 2 a bu u∗ c u2 uu∗ u∗2 β
1 u2 1 u∗2 0
2 (
1 1
1 u u∗
∗2
dx
−
−2 −
−
u
v − v∗ 2 dx
2 1 u2
2 0
1
1
∗ 2 γ
dx.
−
− w − w α 21 u∗2 2
0
3.4
The first integrand in the right hand of the above inequality is positive definite if
4 ∗
w d 2α11 u α12 v α13 wd α21 u 2α22 v α23 wd3 α31 u α32 v 2α33 w
αβ
w∗
w∗
1
1
1
w2
α12 u α21 v
α13 u α31
α23 v βα32
β
α
α
w
w
2
1
1
α23 vw α32 w∗ d 2α11 u α12 v α13 w
>
β
α
2
1
1
α13 uw α31 w∗ d α21 u 2α22 v α23 w
β
α
2
1
1
w∗
α12 u α21 v d3 α31 u α32 v 2α33 w.
α
β
3.5
From the maximum-norm estimate in Theorem 2.1, 3.2 is a sufficient condition of 3.5.
Thus when 3.1 holds, there exists a positive constant δ such that
dHu, v, w
≤ −δ
dt
1
u − u∗ 2 v − v∗ 2 w − w∗ 2 dx.
0
3.6
18
Boundary Value Problems
By integration by parts, Hölder inequality and the maximum-norm estimate in
1
Theorem 2.1, we can see that d/dt 0 u − u∗ 2 v − v∗ 2 w − w∗ 2 dx is bounded from
above. According to Lemma 3.1 in 1 and 3.6, we obtain
|u·, t − u∗ |2 −→ 0,
|v·, t − v∗ |2 −→ 0,
|w·, t − w∗ |2 −→ 0,
t −→ ∞.
3.7
1/2
Using Gagliardo-Nirenberg inequalities, we have |u·, t|∞ ≤ C|u|1/2
1,2 |u|2 . Thus
|u·, t − u∗ |∞ −→ 0,
|v·, t − v∗ |∞ −→ 0,
|w·, t − w∗ |∞ −→ 0,
t −→ ∞.
3.8
That is, u, v, w converges uniformly to E∗ . Since Hu, v, w is decreasing for t > 0, E∗ is
globally asymptotically stable.
Acknowledgments
This work has been partially supported by the China National Natural Science Foundation
no. 10871160, the NSF of Gansu Province no. 096RJZA118, the Scientific Research Fund of
Gansu Provincial Education Department, and the NWNU-KJCXGC-03-47 Foundation.
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