Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 289168, 10 pages
doi:10.1155/2012/289168
Research Article
Uniqueness of Traveling Waves for
a Two-Dimensional Bistable Periodic Lattice
Dynamical System
Chin-Chin Wu
Department of Applied Mathematics, National Chung Hsing University, 250 Kuo Kuang Road,
Taichung 402, Taiwan
Correspondence should be addressed to Chin-Chin Wu, chin@email.nchu.edu.tw
Received 28 December 2011; Accepted 30 January 2012
Academic Editor: Muhammad Aslam Noor
Copyright q 2012 Chin-Chin Wu. 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 traveling waves for a two-dimensional lattice dynamical system with bistable
nonlinearity in periodic media. The existence and the monotonicity in time of traveling waves
can be derived in the same way as the one-dimensional lattice case. In this paper, we derive the
uniqueness of nonzero speed traveling waves by using the comparison principle and the sliding
method.
1. Introduction
In this paper, we study the following two-dimensional 2D lattice dynamical system:
u̇i,j t :
d
ui,j t D2 ui,j t f ui,j t ,
dt
i, j ∈ Z, t > 0,
1.1
where f is a C2 function in R and
D2 ui,j : pi1,j ui1,j pi,j ui−1,j qi,j1 ui,j1 qi,j ui,j−1 − di,j ui,j ,
di,j : pi1,j pi,j qi,j1 qi,j .
1.2
2
Abstract and Applied Analysis
We assume that the coefficients pi,j , qi,j are positive and bounded such that
piN,j pi,j pi,jN ,
qiN,j qi,j qi,jN ,
∀i, j ∈ Z,
1.3
for some positive integer N. Furthermore, we consider the case of bistable nonlinearity, namely,
f0 f1 fa 0,
f 0 < 0 < f a,
f 1 < 0,
1.4
for some constant a ∈ 0, 1. For simplicity, we only consider the case when a ∈ 0, 1/2.
We are interested in planar traveling wave solutions of 1.1 such that
kNr mNs
ui−kN,j−mN t
ui,j t c
∀k, m ∈ Z, t ∈ R,
1.5
for some speed c /
0 and
lim
ui,j t 0,
risj → −∞
lim
ui,j t 1,
risj → ∞
1.6
for any t ∈ R in the direction r, s : cos θ, sin θ for some θ ∈ 0, π/2.
The study of lattice dynamical systems has attracted a lot of attention for past years.
In particular, traveling wave solutions are important due to the wide applications of these
special solutions. For example, the invading of one species to another can be described by
traveling wave solutions see, e.g., 1, 2. The lattice dynamical system arises, for example,
when the habitat is divided into discrete niches in certain biology models. We refer the reader
to, for example, 3–9 for monostable nonlinearity and 10–16 for bistable nonlinearity in a
one-dimensional lattice. In particular, in 16 the authors studied a very general model with
bistable nonlinearity in a 1D lattice. Our purpose of this paper is to extend the result of 16
to the case of multidimensional lattice. For the study of multidimensional lattice, we refer to
17–19. For the simplicity of presentation, we will only consider the 2D lattice dynamical
system 1.1. Our results can be easily extended to the more general case with a convection
term or spatially dependent nonlinearity as in 16.
In a similar manner as that in 16 for 1D lattice case, we can prove the existence
of traveling wave solutions of 1.1–1.6 with profile {Ui,j }i,j∈Z and speed c ∈ R, by
transforming the problem 1.1–1.6 into an integral formulation. Moreover, if the speed
c > 0, then we can obtain U̇i,j t < 0 for all i, j ∈ Z and t ∈ R. We will not repeat the proof here
and focus on the study of the uniqueness of nonzero speed traveling waves. The uniqueness
is in the sense that if there exist two traveling waves with nonzero speeds, then these two
speeds are the same, and two wave profiles are the same except a translation. Due to that the
nonlinearity is independent of spatial variable, our proof of the uniqueness is simpler and
more transparent than that in 16. In fact, motivated by the work of Fife and McLeod 20,
Lemma 3.1 below provides some estimations in terms of a given traveling wave solution
for the solution to the initial value problem for 1.1 with certain initial condition. Moreover,
with Lemma 3.1, we employ the idea of moving coordinate and a sliding method to complete
the proof of uniqueness see Theorem 3.3.
Abstract and Applied Analysis
3
This paper is organized as follows. In Section 2, we give some preliminaries including
a comparison principle. Then we prove uniqueness of traveling wave with nonzero speed in
Section 3.
2. Preliminaries
The following lemma can be easily deduced from 1.5 and 1.6.
Lemma 2.1. Let {ui,j }i,j∈Z be a solution of 1.1–1.6. If c > 0< 0, then ui,j t → 0 → 1 as
t → ∞ and ui,j t → 1 → 0 as t → −∞ for each i, j.
We can determine the sign of the speed c when c / 0 as follows.
Lemma 2.2. Suppose that c /
0, then c has the same sign as −
1
0
fsds.
Proof. For K ≥ max{1, N/|c|}, an integration by parts gives
K
−K
2
u̇i,j t dt K
−K
K
−K
u̇i,j t D2 ui,j t f ui,j t dt
pi1,j u̇i,j ui1,j dt −
K
−K
K
qi,j1 u̇i,j ui,j1 dt −
1
− di,j u2i,j |K
−K 2
K
−K
pi,j ui,j u̇i−1,j dt pi,j ui−1,j ui,j |K
−K
−K
K
−K
2.1
qi,j ui,j u̇i,j−1 dt qi,j ui,j−1 ui,j |K
−K
u̇i,j f ui,j dt for 1 ≤ i, j ≤ N.
Then 1 ≤ ∀j ≤ N,
K
K
N K
2
pN1,j u̇N,j uN1,j dt −
p1,j u̇0,j u1,j dt
u̇i,j t dt i1
−K
−K
N
−K
pi,j ui−1,j ui,j |K
−K i1
−
i1
N K
i1
N K
−K
qi,j ui,j u̇i,j−1 dt −
−K
qi,j1 u̇i,j ui,j1 dt N
1
i1
2
N
qi,j ui,j−1 ui,j |K
−K
i1
di,j u2i,j |K
−K N K
i1
−K
u̇i,j f ui,j dt.
2.2
4
Abstract and Applied Analysis
Therefore,
K
K
N N
N K
2
p1,j
u̇N,j uN1,j dt −
u̇0,j u1,j dt
u̇i,j t dt j1 i1
−K
j1
N
−K
−K
K
K
qi,1
i1
N N j1 i1
−K
−K
u̇i,0 ui,1 dt
1
K
2 K
pi,j ui−1,j ui,j |K
−K qi,j ui,j−1 ui,j |−K − di,j ui,j |−K
2
N K
N j1 i1
u̇i,N ui,N1 dt −
−K
2.3
u̇i,j f ui,j dt.
Sending K to ∞, using 1.5 and Lemma 2.1, it follows that
N ∞
N j1 i1
−∞
2
u̇i,j t dt − sgncN 2
1
fsds.
0
2.4
Hence, the lemma follows.
As a simple consequence of Lemma 2.2, we have c 0 if f is of balanced type, that is,
fsds 0. Notice that we cannot guarantee the speed c is zero or not by using the method
0
developed in 16. In fact, for the 1D lattice case, the classical work of Keener 21 indicates
that the propagation failure i.e., c 0 occurs when the diffusion coefficient is sufficiently
small, even when f is of unbalanced type. A similar result for 1D periodic case can be found
in 22. For our model, the problem for the propagation failure is still open.
Set u : {ui,j }i,j∈Z . Define Ni,j ut : u̇i,j t − D2 ui,j t − fui,j t. Then we have the
following comparison principle.
1
Lemma 2.3. Assume that t0 ∈ R, i0 , j0 ∈ Z ∪ {∞} and c ∈ R. Suppose that u: {ui,j t}i,j∈Z and
v : {vi,j t}i,j∈Z are bounded and continuous on
i, j, t ∈ Z2 × R | t ≥ t0 , ri sj ct ≤ ri0 sj0 1 ,
2.5
such that
Ni,j u ≥ Ni,j v ∀t > t0 , ri sj ct ≤ ri0 sj0 ,
ui,j t0 ≥ vi,j t0 ui,j t ≥ vi,j t
∀ri sj ct0 < ri0 sj0 ,
∀t ≥ t0 , ri0 sj0 ≤ ri sj ct ≤ ri0 sj0 1,
2.6
Abstract and Applied Analysis
5
then ui,j t ≥ vi,j t for all t > t0 , ri sj ct ≤ ri0 sj0 . Moreover, if there exists some i1 , j1 with ri1 sj1 ct0 < ri0 sj0 such that ui1 ,j1 t0 > vi1 ,j1 t0 , then ui,j t > vi,j t for all t > t0 ,
ri sj ct < ri0 sj0 .
Since the proof is quite similar to the one given in 16, Lemma 1, we safely omit it
here.
3. Uniqueness
In this section, we will study the uniqueness of traveling waves of 1.1–1.6. Firstly, applying a method of Fife-McLeod 20, we can derive the following result.
Lemma 3.1. Suppose that u : {ui,j t}i,j∈Z is a solution of 1.1 for t > 0, such that
lim sup ui,j 0 < a < lim inf ui,j 0
3.1
risj → ∞
risj → −∞
c
and 0 < ui,j t < 1 for all t ≥ 0. If Ui,j
t is a traveling wave solution with speed c /
0, then there exist
a sufficiently large positive integer i0 and j0 , depending on the initial value, and positive numbers β0
(depending on the value of f s near s 0 and s 1), δ0 δ0 β0 , l, f, Uc > 0, such that for all
i, j ∈ Z, t > 0,
c
−βt
−βt
c
−βt
t
−
δe
−
le
t
δe
le−βt
≤
u
Ui−i
≤
U
t
i,j
N,
j−j
N
N,
jj
N
ii
0
0
0
0
3.2
if β ∈ 0, β0 , δ ≥ δ0 ,
l ∈ max lim sup ui,j 0, 1 − lim inf ui,j 0 , min{a − , 1 − a − } ,
risj → ∞
risj → −∞
3.3
for some small > 0.
c
Proof. We will only consider the case when c > 0. In this case, we have U̇i,j
t < 0 for all t ∈ R.
First, we let
⎧
⎨ f0 − f p /p, p > 0,
Φ 0, p ⎩ −f 0,
p 0.
3.4
Clearly, Φ0, p is continuous. Fixing with
0 < < min a − lim sup ui,j 0, lim inf ui,j 0 − a ,
risj → −∞
risj → ∞
3.5
there exists μ1 > 0, such that f0−fp/p ≥ 2μ1 for all 0 < p ≤ a−. Since fs is continuous,
we would find Δ1 > 0, such that fu − fu p/p ≥ μ1 for all 0 ≤ u ≤ Δ1 , 0 < p ≤ a − . By
6
Abstract and Applied Analysis
the same reasoning, there exist μ2 > 0 and Δ2 > 0, such that fu − fu p/p ≥ μ2 for all
p ≤ a − , 1 − Δ2 ≤ u ≤ 1.
: {
Define u
ui,j }i,j∈Z , where
c
−βt
le−βt .
u
i,j t : min 1, Ui
i N, jj N t δe
0
0
3.6
Choose l satisfying lim suprisj → −∞ ui,j 0 < l < a − < a, and let i0 , j0 be determined later. We
claim that
c
ui,j 0 ≤ Ui
i N, jj N δ l,
0
∀i, j ∈ Z,
0
3.7
for some i0 and j0 . Since lim suprisj → −∞ ui,j 0 < l, there exists m, such that
ui,j 0 < l,
3.8
∀ri sj ≤ m.
Moreover, since limrisj → ∞ Ui,j 0 1, there exist i0 and j0 , such that
c
ui,j 0 ≤ Ui
i N, jj N δ l,
0
∀ri sj ≥ m.
0
3.9
Combining 3.8 and 3.9, we have proved the claim 3.7.
−βt
≥ 0. If u
i,j t Uc t δe
le−βt , then
Now, we prove Ni,j u
ii0 N, jj0 N
c
−βt
−βt
−βt
1 − δβe
Ui
t
δe
− l βe
Ni,j u
i N, jj N
0
0
c
c
−βt
−βt
le−βt − f Ui
le−βt
− D2 Ui
i N, jj N t δe
i N, jj N t δe
0
0
0
−βt Uc
−δβe
ii N, jj N
0
0
−βt − l βe
−βt
t δe
c
−βt
f Ui
i N, jj N t δe
0
0
0
3.10
c
−βt
le−βt
− f Ui
t
δe
.
i N, jj N
0
0
Divide into three cases.
c
Case 1. 0 ≤ Ui,j
≤ Δ : min{Δ1 , Δ2 }. From the above discussion, since le−βt ≤ l < a − , we
have
c
c
f Ui,j
− f Ui,j
le−βt ≥ μ1 le−βt .
3.11
c
−βt μ1 le−βt μ1 − β
le−βt . Choosing β ≤ μ1 , we have Ni,j u
≥ −l βe
≥ 0 in
Since U̇i,j
< 0, Ni,j u
this case.
Abstract and Applied Analysis
7
c
−βt
≥ −l βe
Case 2. 1 − Δ ≤ Ui,j
≤ 1. As in Case 1, we have μ2 > 0 such that Ni,j u
μ2 le−βt le−βt
μ2 − β
≥ 0 if β ≤ μ2 .
c
c
Case 3. Δ ≤ Ui,j
≤ 1 − Δ. If σ : minΔ≤Ui,jc ≤1−Δ |U̇i,j
| > 0 and k : maxs∈0,1 |f s| > 0, then
−βt
−βt
− lβ − kle−βt
we have
≥ δβσe
Ni,j u
− lβe
− kle−βt δβσ
. Choosing δ ≥ lβ k/βσ,
≥ 0 in this case.
Ni,j u
≥ 0 for all cases. Hence, the second inequality of 3.2 follows from a
Then Ni,j u
comparison principle. By the same way, we have the first inequality of 3.2. This proves
the lemma.
Note that we have the following different type of super- and sub-solutions which can
be verified by a similar way as that of Lemma 3.1.
Lemma 3.2. Suppose that w : {wi,j t}i,j∈Z is a solution of 1.1–1.6. For any t0 ∈ R, l ∈ 0, l0 ,
c
l0 : min{a − , 1 − a − }, is a small number, and σ1 ≥ σ0 σ0 β, f, wi,j
> 0, and let w± ±
{wi,j
t}
, where
i,j∈Z
±
wi,j
t : wi,j t t0 ∓ σ1 l 1 − e−βt ± le−βt ,
3.12
then ±Ni,j w± ≥ 0.
We now prove the following uniqueness result.
c
c
Theorem 3.3. Suppose that {Ui,j
}, {Ui,j } are two traveling wave solutions of 1.1–1.6 with c,
c
c
c
/ 0, then one has c c and Ui,j ξ Ui,j ξ ξ∗ for some ξ∗ ∈ R.
Proof. As before, we only consider the case when both c and c are positive. Since
lim
Ui,j 0 0 < a <
risj → −∞
lim Ui,j 0 1,
risj → ∞
3.13
by Lemma 3.1, we have
c
c
−βt
−βt
c
−βt
U
t
−
δe
−
le
t
δe
le−βt ,
Ui−i
≤
≤
U
t
i,j
N,
j−j
N
N,
jj
N
ii
0
0
0
0
3.14
for all t > 0, i, j ∈ Z with the constants i0 , j0 , l, β, δ defined in Lemma 3.1.
Let Ik I kN, Jm J mN, and I, J ∈ {1, . . . , N}, k, m ∈ Z. Take i, j Ik , Jm . We
get for all t > 0, k, m ∈ Z,
c
c
−βt
UIkN−i
t
−
δe
− le−βt ≤ UIkN, JmN t
0 N, JmN−j0 N
c
t δe−βt le−βt ,
≤ UIkNi
0 N, JmNj0 N
∀I, J.
3.15
8
Abstract and Applied Analysis
By the property 1.5 for all t > 0, k, m ∈ Z,
c
rk smN
rk smN
−βt
−βt
− δe
t−
− le ≤ UI,J t −
c
c
rk smN
c
≤ UIi
δe−βt le−βt , ∀I, J.
t−
0 N, Jj0 N
c
c
UI−i
0 N, J−j0 N
3.16
Setting the moving coordinate
ξ : t −
rk smN
,
c
3.17
we have for any ξ ∈ R, for all I, J,
N N
c
−βt
−
δe
−
UI−i
ξ
sm
− le−βt
rk
0 N,J−j0 N
c
c
c
3.18
≤ UI,J ξ
N N
c
−βt
−
ξ
sm
le−βt .
≤ UIi
δe
rk
0 N,Jj0 N
c
c
Suppose that c /
c. We may assume that c < c. Fixing ξ and sending t → ∞, this leads that
c
c
either UI,J ξ ≡ 0 or UI,J ξ ≡ 1, which is a contradiction. Hence, c c.
We now suppress the dependence of c, and we obtain
UI−i0 N, J−j0 N ξ ≤ UI,J ξ ≤ UIi0 N, Jj0 N ξ,
∀ξ ∈ R, ∀I, J.
3.19
For ξ0 : ri0 sj0 N/c, we have
UI,J ξ ξ0 ≤ UI,J ξ ≤ UI,J ξ − ξ0 ,
∀ξ ∈ R, ∀I, J.
3.20
Define
ξ∗ : inf ξ | UI,J ξ ξ ≤ UI,J ξ, ∀ξ ∈ R, ∀I, J ,
ξ : sup ξ | UI,J ξ ≤ UI,J ξ ξ , ∀ξ ∈ R, ∀I, J .
∗
3.21
Since U̇i,j t < 0 due to c > 0, we have ξ∗ ≤ ξ∗ . Assume that ξ∗ < ξ∗ . By the strong comparison
principle, we know that
UI,J ξ < UI,J ξ ξ∗ ,
∀ξ ∈ R, ∀I, J.
3.22
Abstract and Applied Analysis
9
Since lim|ξ| → ∞ U̇I,J ξ 0, there exists M > 0 such that
2σ1 U̇I,J ξ ≤ 1,
∀|ξ | ≥ M, ∀I, J,
3.23
where σ1 is the number mentioned in Lemma 3.2. If |ξ ξ∗ | ≤ M 1, by the continuity of UI,J ,
we would find h ∈ 0, η, η : min{1/2σ1 , l0 }, such that
UI,J ξ < UI,J ξ ξ∗ 2σ1 h,
3.24
∀I, J.
If |ξ ξ∗ | ≥ M 1, then
UI,J ξ ξ∗ 2σ1 h − UI,J ξ > UI,J ξ ξ∗ 2σ1 h − UI,J ξ ξ∗ U̇I,J ξ ξ∗ θ2σ1 h2σ1 h,
≥ −h,
θ ∈ 0, 1
3.25
∀I, J.
Combining 3.24 and 3.25, we get
UI,J ξ ≤ UI,J ξ ξ∗ 2σ1 h h,
∀ξ ∈ R, ∀I, J.
3.26
Hence,
UI,J 0 ≤ UI,J ξ∗ 2σ1 h h,
3.27
∀I, J.
By Lemma 3.2 and the comparison principle, we have
UI,J ξ ≤ UI,J ξ ξ∗ 2σ1 h − σ1 h 1 − e−βt he−βt ,
∀ξ ∈ R, ∀I, J.
3.28
Fixing ξ and sending t → ∞,
UI,J ξ ≤ UI,J ξ ξ∗ σ1 h.
3.29
This contradicts with the definition of ξ∗ . Hence, Ui,j ξ Ui,j ξ ξ∗ .
Hence, we obtain the uniqueness up to translations of the traveling wave solution
with nonzero speed.
Acknowledgments
This work was partially supported by the National Science Council of the Republic of China
under the Grant NSC 100-2115-M-005-001. The authors also thank the referee for some
valuable comments and suggestions.
10
Abstract and Applied Analysis
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Volume 2014
Journal of
Complex Analysis
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Volume 2014
International
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Stochastic Analysis
Abstract and
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International Journal of
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Volume 2014
Discrete Dynamics in
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Volume 2014
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Optimization
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Volume 2014
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Volume 2014