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Nonlinear stability of traveling wave fronts for nonlocal delayed reaction–diffusion equations
J. Math. Anal. Appl. 385 (2012) 1094–1106
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Nonlinear stability of traveling wave fronts for nonlocal delayed reaction–diffusion equations ✩ Guangying Lv a,∗ , Mingxin Wang b a b
College of Mathematics and Information Science, Henan University, Kaifeng 475001, PR China Science Research Center, Harbin Institute of Technology, Harbin 150080, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 28 January 2011 Available online 23 July 2011 Submitted by J. Shi
This paper is concerned with the nonlinear stability of traveling wave fronts for nonlocal delayed reaction–diffusion equation. We prove that these traveling wave fronts are exponentially stable to perturbation in some exponentially weighted L ∞ spaces, when the initial perturbation around the traveling wave fronts decays exponentially as x → −∞, but the initial perturbation can be arbitrary large in other locations. The time decay rate is also obtained by weighted energy estimates. © 2011 Elsevier Inc. All rights reserved.
Keywords: Delay Stability Nonlocal reaction–diffusion equation Traveling wave fronts Weighted energy
1. Introduction In this paper, we study nonlinear stability of traveling wave fronts for the following nonlocal delayed reaction–diffusion equation
ut = J ∗ u − u + f u (x, t ), u (x, t − τ ) ,
x ∈ R, t > 0,
τ 0, J ∈ C 1 (R) is a nonnegative even function and satisfies J ( y ) d y = 1, J ∗u= J ( x − y )u ( y , t ) d y , J ( y )eλ y d y < ∞
(1.1)
where constant
R
R
(1.2)
R
for all λ ∈ R. It is well known that there exists a unique (up to translation) wave front of Eq. (1.1) when τ = 0, see Bates et al. [3], Carr and Chamj [4], Chen [5] and references therein. The asymptotic stability of traveling wave fronts for Eq. (1.1) with τ = 0 was obtained by Chen [5] using squeezing method. Bates and Chen [1] established the multidimensional stability of traveling wave fronts for Eq. (1.1) with τ = 0 using spectral analysis. Recently, Pan, Li and Lin [18] considered Eq. (1.1) under the condition that function f satisfies quasi-monotonicity and established the existence of traveling wave fronts by using Schauder’s fixed point theorem and upper–lower solution technique. In this paper, we are interested in the nonlinear stability of traveling wave fronts for Eq. (1.1). ✩ This work was supported by PRC Grants NSFC 11071049, JSPS Innovation Program CX09B_044Z and the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1009). Corresponding author. E-mail addresses: [email protected] (G.Y. Lv), [email protected] (M.X. Wang).
*
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.07.033
©
2011 Elsevier Inc. All rights reserved.
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
1095
Traveling wave fronts of delayed reaction–diffusion equations have been studied by many authors. Schaaf [20] studied two scalar reaction–diffusion equations with a discrete delay for both Huxley nonlinearity and Fisher nonlinearity and established the existence of traveling wave fronts and uniqueness of wave speeds by a phase plane analysis method. Wu and Zou [24] considered a more general reaction–diffusion system with finite delay and obtained the existence of traveling wave fronts by using the classical monotone iteration technique with sub-supersolution method. Using the method of Wu and Zou [24], Zou [25] obtained the existence of traveling wave fronts for the delayed KPP equation, and recently, Lin and Hong [9] extends the results of [25] to a host-vector disease model
ut = du xx − au (x, t ) + bu (x, t − τ ) 1 − u (x, t ) ,
x ∈ R, t > 0,
where constants b > a 0. The study of uniqueness and asymptotic stability of traveling wave fronts become relatively more difficult. Sattinger [19] studied a reaction–diffusion system without delay. By detail spectral analysis, he proved that the traveling wave fronts were stable to perturbations in some exponentially weighted L ∞ spaces. Smith and Zhao [21] considered Eq. (1.1) with J ∗ u − u replaced by u. They first established the existence and comparison theorem of solution for a quasi-monotone reaction–diffusion bistable equation with a discrete delay and then obtained the stability of traveling wave fronts by using the elementary super-subsolution comparison and squeezing methods developed by Chen [5] (see also [6,22,23] for this technique). Just recently, Mei et al. [12] considered the so-called Nicholson’s blowflies equation with diffusion
v t − D m v xx + dm v = εb v (x, t − r ) , where constants D m > 0, dm > 0,
b1 ( v ) = p ve−av
q
(1.3)
ε > 0 and pv
or b2 ( v ) =
1 + av q
(1.4)
.
They first established a comparison principle and then proved that traveling wave fronts of Eq. (1.3) are asymptotic stable in some exponentially weighted L ∞ spaces. Lv and Wang [11] considered some more general models and established the stability of traveling wave fronts using the method developed by Mei et al. [12]. Meanwhile, Murray [17] pointed out that the general reaction–diffusion equations
ut = du xx + f (u ),
x ∈ R, t > 0
are strictly only applicable to dilute systems and population models. Moreover, time delay seems unavoidable in a real world. Thus Eq. (1.1) is a more suitable model. To our best knowledge, there is no result about the stability of traveling wave fronts for Eq. (1.1). Encouraged by papers [12,15], in this paper we study the nonlinear stability of traveling wave fronts for Eq. (1.1) with f = −au + bu (x, t − τ )(1 − u ) and a < b. The corresponding Cauchy problem is
ut = J ∗ u − u − au + bu (x, t − τ )(1 − u ), u (x, s) = u 0 (x, s),
x ∈ R, t > 0,
(1.5)
x ∈ R, s ∈ [−τ , 0].
(1.6)
It is remarked that there is some difference between the problem (1.5)–(1.6) and the following problem
ut = u xx + f u (x, t ), u (x, t − τ ) ,
x ∈ R, t > 0,
u (x, s) = u 0 (x, s),
x ∈ R, s ∈ [−τ , 0].
(1.7)
It follows from the basic estimates of paper [11] that the solution u (x, t ) of (1.7) satisfies
0 2 2 2 u ξ (s) 2 C u 0 (0) 2 + u 0 (s) 2 ds . L L L w
w
w
−τ
However, in the basic estimates of solution to the problem (1.5)–(1.6) doesn’t contain the above term, we cannot obtain the derivative estimate by the method of [11]. We shall use Young-inequality and the behavior of function u to overcome the difficulty. Throughout this paper, C > 0 denotes a generic constant, while C i (i = 1, 2, . . .) represents a specific constant. Let I be an interval, typically I = R. Denote by L 2 ( I ) the space of square integrable functions defined on I , and H k ( I ) (k 0) the Sobolev space of the L 2 -functions f (x) defined on the interval I whose derivatives Let
L 2w ( I )
2
di dxi
f (i = 1, . . . , k) also belong to L 2 ( I ).
be the weighted L -space with a weight function w (x) > 0 and its norm is defined by
f L 2w ( I ) = I
2
12
w (x) f (x) dx
.
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G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
Let H kw ( I ) be the weighted Sobolev space with the norm given by
f Hk (I ) = w
k i =0 I
12
i
d f (x) 2
dx . w (x)
i
dx
Let T > 0 be a number and B be a Banach space. We denote by C 0 ([0, T ]; B ) the space of the B-valued continuous function on [0, T ], and by L 2 ([0, T ]; B ) the space of the B-valued L 2 -functions on [0, T ]. The corresponding spaces of the B-valued L 2 -functions on [0, ∞) are defined similarly. The rest of this paper is organized as follows. In Section 2, we introduce the traveling wave fronts and state the stability theorems. Section 3 is concerned with the proof of the stability theorem. In Section 4, the nonlinear stability of traveling wave fronts of other models with or without delay are obtained by using the similar method. This paper ends with a short discussion. 2. Preliminaries and main theorem In this section, we first recall some known results, then define weight function and last state our main theorem. Since a < b, it is easy to see that Eq. (1.5) has two nonnegative constant equilibria u − = 0 and u + = 1 − a/b. A traveling wave front of Eq. (1.5) connecting with the constant states u ± is a solution u with the form u (x, t ) = φ(x + ct ) and satisfies φ (ξ ) 0 and
c φ (ξ ) = ( J ∗ φ)(ξ ) − φ(ξ ) − aφ(ξ ) + bφ(ξ − c τ ) 1 − φ(ξ ) ,
(2.1)
φ(±∞) = u ± ,
where ξ = x + ct, = ddξ . For the special case a = 0, the existence of traveling wave front has been obtained by the authors of [18]. When the problem (2.1) has a pair of upper and lower solutions, it follows from [18] that the problem (2.1) has at least one solution. Let
¯ ) = min 1 − a/b, eλ1 (c )ξ , φ(ξ where q > 1 is large enough, equation
φ(ξ ) = max 0, eλ1 (c )ξ − qeκ λ1 (c )ξ ,
κ ∈ (1, min{2, λ2 (c )/λ1 (c )}), and λ1 (c ) and λ2 (c ) are two positive roots of the following
c (λ) = J ∗ eλ· − 1 − c λ − a + be−λc τ ,
(2.2)
and satisfy λ1 (c ) < λ2 (c ), where
J ( y )eλ y d y .
J ∗ eλ· = R
¯ ) and φ(ξ ) are a pair The existence of λ1 (c ), λ2 (c ) will be given in the following Proposition 2.1. It is easy to verify that φ(ξ of upper and lower solutions of Eq. (2.1). Proposition 2.1. Let c (λ) define as (2.2), then there exist c ∗ > 0 and λ∗ such that c (λ) = 0 has two distinct positive roots λ2 (c ) > λ∗ > λ1 (c ) for each c > c ∗ and one positive root λ∗ for c = c ∗ . And for each c c ∗ , Eq. (1.5) has a traveling wave front φ(x + ct ) satisfying φ(±∞) = u ± . Moreover, if c > c ∗ ,
lim φ(ξ )e−λ1 (c )ξ = 1,
ξ →−∞
lim φ (ξ )e−λ1 (c )ξ = λ1 (c ).
ξ →−∞
The proof of Proposition 2.1 is standard and we omit it. Throughout this paper, we always assume that there exists a positive constant
+∞ J ( y )eη y d y <
1 2
2
+ (b − a). 3
η such that (2.3)
0
We define a weight function as
w (ξ ) =
e−η(ξ −ξ0 )
for ξ ξ0 ,
1
for ξ > ξ0 ,
−a) where ξ0 is chosen to be large enough such that φ(ξ0 − c τ ) > 2(b3b . Now, we state our main theorem.
(2.4)
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
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Theorem 2.1. Let
c¯ =
1
2b − 2a −
η
1
∗ J ( y )eη y d y + J ∗ eλ · .
+
2
R
For any given traveling wave front φ(x + ct ) of (1.5) with speed c > max{¯c , c ∗ } and τ > 0, if the initial data satisfies
u − u 0 (x, s) u +
for (x, s) ∈ R × [−τ , 0],
and the initial perturbation u 0 (x, s) − φ(x + cs) belongs to C ([−τ , 0], H 1w (R)), then the solution of (1.5)–(1.6) satisfies
u − u (x, t ) u +
for (x, t ) ∈ R × R+ ,
u (x, t ) − φ(x + ct ) ∈ C [0, +∞); H 1w (R) , where the function w (x) is defined by (2.4). Moreover, the solution u (x, t ) converges to the traveling wave front φ(x + ct ) exponentially in time:
sup u (x, t ) − φ(x + ct ) C e−μt x∈R
for some positive constants C and μ. Remark 2.1. It is easy to show that the results of Theorem 2.1 also hold for a = 0 and b > 0. In other words, we also establish the stability of traveling wave fronts for nonlocal delayed KPP equation. Remark 2.2. It follows from the assumptions of Theorem 2.1 that know that
c η > c¯ η = 2b − 2a −
= J ∗e
λ∗ ·
> J ∗e
λ1 (c )·
1 2
+
η > λ1 (c ). In fact, from the choice of the speed c, we
∗ J ( y )eη y d y + J ∗ eλ ·
R
− 1 − a + be
−λ1 (c )c τ
− 1 − a + be
+
1 2
+b−a+b 1−e
−λ1 (c )c τ
J ( y )eη y d y
+ R
−λ1 (c )c τ
= c λ1 (c ), ∗ where we have used the facts that b > a, e−λ1 (c )c τ < 1, and J ∗ eλ · J ∗ eλ1 (c )· . In fact, direct calculations show that
J ∗e
λ·
λ
=
∞ J ( y ) ye
λy
dy =
R
J ( y ) y eλ y − e−λ y d y 0,
λ > 0.
0
Hence we have that η > λ1 (c ). We also remark that the choice of the speed c depends on we first choose the positive constant η satisfying (2.3), and then we define the speed c.
η. Actually, for any fixed b > a 0,
3. Proof of Theorem 2.1 In this section, we first establish a comparison principle for the problem (1.5)–(1.6) and then prove Theorem 2.1 by using the weighted energy method. The method we used here is similar to Mei et al. [12] and Lv and Wang [11]. Lemma 3.1. Let T > 0 and Q T = R × (0, T ]. Assume that the nonnegative function c (x, t ) is bounded for (x, t ) ∈ Q T . If the function u satisfies
ut − J ∗ u + u + c (x, t )u 0 ( 0),
(x, t ) ∈ Q T ,
u (x, 0) 0 ( 0),
x ∈ R,
then u (x, t ) 0 ( 0) for (x, t ) ∈ Q T . The proof of this lemma is easy and we omit it.
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G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
Lemma 3.2. Let
u − = 0 u 0 (x, s) u +
for (x, s) ∈ R × [−τ , 0].
Then the solution u (x, t ) of the Cauchy problem (1.5)–(1.6) satisfies
u − u (x, t ) u +
for (x, t ) ∈ R × R+ .
Proof. We first prove u (x, t ) 0. For t ∈ [0, τ ], we have u (x, t − τ ) = u 0 (x, t − τ ). Thus u (x, t ) satisfies the following equation
ut − J ∗ u + u + a + bu 0 (x, t − τ ) u = bu 0 (x, t − τ ) 0,
(x, t ) ∈ R × [0, τ ],
u (x, s) = u 0 (x, s) 0,
(x, s) ∈ R × [−τ , 0].
It follows from Lemma 3.1 that u (x, t ) 0 on R × [0, τ ]. Repeating this procedure to each of the intervals [nτ , (n + 1)τ ], n = 1, 2, . . . , we have u (x, t ) 0 on R × R+ . Next, we prove u (x, t ) u + . It is easy to see that
(u − u + )t − J ∗ (u − u + ) + u − u + + a + bu (x, t − τ ) (u − u + ) = a u (x, t − τ ) − u + , (x, t ) ∈ R × R+ , u (x, s) − u + = u 0 (x, s) − u + ,
(x, s) ∈ R × [−τ , 0].
Similar to the proof of u (x, t ) 0, we first consider t ∈ [0, τ ]. Note that 0 u 0 (x, s) u + on R × [−τ , 0], we have
⎧ ⎪ ⎨ (u − u + )t − J ∗ (u − u + ) + u − u + + a + bu 0 (x, t − τ ) (u − u + ) (x, t ) ∈ R × [0, τ ], = a u 0 (x, t − τ ) − u + 0, ⎪ ⎩ u (x, s) − u + = u 0 (x, s) − u + 0, (x, s) ∈ R × [−τ , 0].
Again by Lemma 3.1, we have u (x, t ) u + on R × [0, τ ]. Repeating the above procedure to each of the intervals [nτ , (n + 1)τ ], n = 1, 2, . . . , we have u (x, t ) u + on R × R+ . This completes the proof. 2 + Lemma 3.3 (Comparison principle). Let u − (x, t ) and u + (x, t ) be solutions of Eq. (1.5) with initial data u − 0 (x, s) and u 0 (x, s), respectively. If
+ u− u− 0 (x, s) u 0 (x, s) u +
for (x, s) ∈ R × [−τ , 0],
u − u − (x, t ) u + (x, t ) u +
for (x, t ) ∈ R × R+ .
then
Proof. It follows from Lemma 3.2 that
u − u − (x, t ) u + ,
u − u + (x, t ) u +
for (x, t ) ∈ R × R+ .
− Now we prove u − (x, t ) u + (x, t ). Let u (x, t ) = u + (x, t ) − u − (x, t ) and u 0 (x, s) = u + 0 (x, s) − u 0 (x, s), then u (x, t ) and u 0 (x, s) satisfy
ut − J ∗ u + u + a + bu + (x, t − τ ) u = bu (x, t − τ ) 1 − u − , (x, t ) ∈ R × [0, τ ], u (x, s) = u 0 (x, s),
(x, s) ∈ R × [−τ , 0].
From the assumptions, we know that u 0 (x, s) 0 on R × [−τ , 0]. Noting that u − (x, t ) u + = 1 −
− 0, (x, t ) ∈ R × [0, τ ], ut − J ∗ u + u + a + bu + 0 (x, t − τ ) u = bu 0 (x, t − τ ) 1 − u
u (x, s) = u 0 (x, s) 0,
a b
< 1, we have
(x, s) ∈ R × [−τ , 0].
Hence by Lemma 3.1, we have u (x, t ) 0 on R × [0, τ ], that is, u − (x, t ) u + (x, t ) on R × [0, τ ]. Repeating the above procedure to each of the intervals [nτ , (n + 1)τ ], n = 1, 2, . . . , we have u − (x, t ) u + (x, t ) on R × R+ . This completes the proof.
2
Following the idea of [12,15], we shall use the comparison principle and weighted energy method to prove Theorem 2.1. Let the initial data u 0 (x, s) satisfies
u − = 0 u 0 (x, s) u + Define
for (x, s) ∈ R × [−τ , 0].
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
1099
u− 0 (x, s) = min u 0 (x, s), φ(x + cs) ,
(x, s) ∈ R × [−τ , 0], u 0 (x, s) = max u 0 (x, s), φ(x + cs) , (x, s) ∈ R × [−τ , 0].
+
Obviously, + u− u− 0 (x, s) u 0 (x, s) u 0 (x, s) u + , −
(x, s) ∈ R × [−τ , 0],
+
u − u 0 (x, s) φ(x + cs) u 0 (x, s) u + ,
(3.1)
(x, s) ∈ R × [−τ , 0].
(3.2)
+ Let u − (x, t ) and u + (x, t ) be the solutions of Eqs. (1.5)–(1.6) with initial data u − 0 (x, s) and u 0 (x, s), respectively. It follows
comparison principle that
u − u − (x, t ) u (x, t ) u + (x, t ) u + ,
(x, s) ∈ R × R+ ,
u − u − (x, t ) φ(x + ct ) u + (x, t ) u + ,
(3.3)
(x, s) ∈ R × R+ .
(3.4)
Proof of Theorem 2.1. We prove Theorem 2.1 in three steps. Step 1. We first prove the convergence of u + (x, t ) to φ(x + ct ). Let ξ = x + ct and
u 0 (ξ, s) = u + 0 (x, s) − φ(x + cs).
u (ξ, t ) = u + (x, t ) − φ(x + ct ), Then by (3.2) and (3.4), we have
u (ξ, t ) 0,
u 0 (ξ, s) 0.
Moreover, u (ξ, t ) satisfies
ut + cu ξ − J ∗ u + u + a + bφ(ξ − c τ ) u = bu (ξ − c τ , t − τ )(1 − u − φ),
(ξ, t ) ∈ R × R+ ,
u (ξ, s) = u 0 (ξ, s),
(ξ, s) ∈ R × [−τ , 0].
Let w (ξ ) > 0 be the weight function defined in (2.4). Multiplying (3.5) by e2μt w (ξ )u (ξ, t ) where later in Lemma 3.5, we have
1 2
e2μt wu 2
+ t
c 2
e2μt wu 2
(3.5)
μ > 0 will be specified
− e2μt w (ξ )u (ξ, t ) ξ
J ( y )u (ξ − y , t ) d y R
c w + − − μ + 1 + a + bφ(ξ − c τ ) wu 2 e2μt 2 w
= b(1 − φ) wuu (ξ − c τ , t − τ )e2μt − bu (ξ − c τ , t − τ )u 2 we2μt .
(3.6)
Integrating (3.6) with respect to (ξ, t ) over R × [0, t ] and dropping the negative term
t bu (ξ − c τ , s − τ )u 2 (ξ, s) w (ξ )e2μs dξ ds,
− 0 R
we obtain
2
e2μt u (t ) L 2 − 2
t e2μs w (ξ )u (ξ, s)
w
0 R
t +
J ( y )u (ξ − y , s) d y dξ ds R
w e2μs −c − 2μ + 2 + 2a + 2bφ(ξ − c τ ) w (ξ )u 2 (ξ, s) dξ ds w
0 R
2 u 0 (0) L 2 + 2b
t
e2μs 1 − φ(ξ ) w (ξ )u (ξ, s)u (ξ − c τ , s − τ ) dξ ds.
w
0 R
By using the Cauchy–Schwarz inequality 2xy x2 + y 2 , we can estimate that
(3.7)
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G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
t e2μs w (ξ )u (ξ, s)
2 0 R
J ( y )u (ξ − y , s) d y dξ ds R
t
t e2μs w (ξ )u 2 (ξ, s) dξ ds +
e2μs w (ξ )
0 R
0 R
t
t e2μs w (ξ )u 2 (ξ, s) dξ ds +
= 0 R
J ( y )u 2 (ξ − y , s) d y dξ ds R
e2μs w (ξ )u 2 (ξ, s)
0 R
J ( y)
w (ξ + y )
R
w (ξ )
d y dξ ds
(3.8)
and
t
e2μs 1 − φ(ξ ) w (ξ )u (ξ, s)u (ξ − c τ , s − τ ) dξ ds
2b 0 R
t
e2μs 1 − φ(ξ ) w (ξ ) u 2 (ξ, s) + u 2 (ξ − c τ , s − τ ) dξ ds
b 0 R
t =b
e
2μ s
1 − φ(ξ ) w (ξ )u 2 (ξ, s) dξ ds + be2μτ
e2μs 1 − φ(ξ + c τ ) w (ξ + c τ )u 2 (ξ, s) dξ ds −τ R
0 R
t b
t −τ
e
2μ s
1 − φ(ξ ) w (ξ )u 2 (ξ, s) dξ ds + be2μτ
0 R
t
e2μs 1 − φ(ξ + c τ ) w (ξ + c τ )u 2 (ξ, s) dξ ds 0 R
0
+ be2μτ
e2μs 1 − φ(ξ + c τ ) w (ξ + c τ )u 20 (ξ, s) dξ ds.
(3.9)
−τ R
Submitting (3.8) and (3.9) into (3.7) and using 0 φ(ξ ) 1 and
e
2μt
2 u (t ) L 2 +
μ ∈ (0, μ1 ) (see Lemma 3.4), we have
t e2μs B μ, w (ξ ) w (ξ )u 2 (ξ, s) dξ ds
w
0 R
0
2 u 0 (0) L 2 + be2μτ
−τ R
C1
e2μs 1 − φ(ξ + c τ ) w (ξ + c τ )u 2 (ξ, s) dξ ds
w
u 0 (0)22 + L
0
w
u 0 (s)22 ds , L
(3.10)
w
−τ
where
w (ξ + c τ ) , w (ξ ) w (ξ + y ) w (ξ ) + 1 + 2a + 2bφ(ξ − c τ ) − A w (ξ ) = −c J ( y) dy w (ξ ) w (ξ )
B μ, w (ξ ) = A w (ξ ) − 2μ − b e2μτ − 1 1 − φ(ξ + c τ )
R
w (ξ + c τ ) . − b 1 − φ(ξ ) − b 1 − φ(ξ + c τ ) w (ξ ) In order to get the basic estimate, we must prove B μ, w (ξ ) C > 0 for some constant C . For this aim we need the following key lemma. Lemma 3.4. Let c > c¯ , then there exists positive constant C 2 such that
A w (ξ ) C 2 ,
ξ ∈ R.
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
Proof. Case 1: ξ ξ0 . It follows from (2.4) that
A w (ξ ) = −c
w (ξ ) w (ξ )
w (ξ +c τ ) w (ξ )
1. Noting that 0 φ(ξ ) 1 − a/b and J (− y ) = J ( y ), we have
+ 1 + 2a + 2bφ(ξ − c τ ) −
J ( y)
w (ξ + y ) w (ξ )
R
dy
w (ξ + c τ ) − b 1 − φ(ξ ) − b 1 − φ(ξ + c τ ) w (ξ )
= c η + 1 + 2a + 2bφ(ξ − c τ ) − eη(ξ −ξ0 ) −
ξ0 −ξ
J ( y ) e−η y − eη(ξ −ξ0 ) d y
−∞
w (ξ + c τ ) − b 1 − φ(ξ ) − b 1 − φ(ξ + c τ ) w (ξ ) 1 J ( y )eη y d y − 2b c η + + 2a − 2
R
:= C 3 > 0, where we have used c > c¯ . Case 2: ξ > ξ0 . In this case, w (ξ ) = w (ξ + c τ ) = 1. Since φ(ξ ) is increasing on R and J (− y ) = J ( y ), we obtain ξ0 −ξ
A w (ξ ) = 2a + 2bφ(ξ − c τ ) −
J ( y ) e−η y − 1 d y − b 1 − φ(ξ ) − b 1 − φ(ξ + c τ )
−∞
0 2bφ(ξ0 − c τ ) − b + a −
J ( y ) e− η y − 1 d y
−∞
0 = 2bφ(ξ0 − c τ ) − b + a −
J ( y )e−η y d y +
−∞
1 2
:= C 4 > 0 by the condition (2.3). Finally, let C 2 := min{C 3 , C 4 }. Then we have A η, w (ξ ) C 2 > 0.
2
Lemma 3.5. Let μ1 > 0 be the unique root of the following equation
C 2 /2 − 2μ1 − b e2μ1 τ − 1 = 0. Then B μ, w (ξ ) C 5 := C 2 /2 > 0 on R, for 0 < μ μ1 . Proof. Note that 0 φ(ξ ) 1 on R and
w (ξ + c τ ) w (ξ )
⎧ −η c τ < 1 ⎪ for ξ ξ0 − c τ , ⎨e = eη(ξ −ξ0 ) 1 for ξ0 − c τ < ξ ξ0 , ⎪ ⎩ 1 for ξ > ξ0 .
It is easy to see that, for 0 < μ μ1 ,
w (ξ + c τ ) w (ξ )
B η,μ, w (ξ ) = A η, w (ξ ) − 2μ − b e2μτ − 1 1 − φ(ξ + c τ )
C 2 /2 − 2μ − b e2μτ − 1 := C 5 > 0. The proof is completed.
2
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Submitting this result into (3.10), we have
e
2μt
2 u (t ) L 2 + C 5
t e
w
2μ s
0 2 2 2 u (s) L 2 ds C 1 u 0 (0) L 2 + u 0 (s) L 2 ds . w
w
−τ
0
t
Dropping the positive terms
e
2μt
(3.11)
w
0
e2μs u (s)22 ds, we obtain the basic estimate Lw
0 2 2 2 u (t ) L 2 C 1 u 0 (0) L 2 + u 0 (s) L 2 ds . w
w
(3.12)
w
−τ
Next, we differentiate (3.5) with respect to ξ , and then multiply the resulting equation by e2μt w (ξ )u ξ (ξ, t ) where w (ξ ) again denotes the function in (2.4), with the same value for ξ0 , we obtain
1 2
e2μt wu 2ξ
+ t
+ −c
w 2w
c 2
e2μt wu 2ξ
− e2μt w (ξ )u ξ (ξ, t ) ξ
J ( y )u ξ (ξ − y , t ) d y R
− μ + 1 + a + bφ(ξ − c τ ) wu 2ξ e2μt
= b(1 − φ − u ) wu ξ u ξ (ξ − c τ , t − τ )e2μt − bu (ξ − c τ , t − τ )u 2ξ we2μt − bφξ (ξ − c τ )uu ξ we2μt − bφξ u ξ we2μt u (ξ − c τ , t − τ ). Integrating over R × [0, t ] and carrying out similar steps to those that led to (3.11), we obtain
e
2μt
2 u ξ (t ) L 2 + C 5
t
2
e2μs u ξ (s) L 2 ds
w
w
0
u 0ξ (0)22 + L
C1
0
w
u 0ξ (s)22 ds L
w
−τ
t −b
φξ (ξ − c τ )u (ξ, s) + φξ (ξ )u (ξ − c τ , s − τ ) e2μs u ξ (ξ, s) w (ξ ) dξ ds.
(3.13)
0 R
It is remarked that there is a little difference in getting (3.11) and (3.13). It is easy to see that 1 − φ(ζ ) will be replaced by 1 − u (ζ, t ) − φ(ζ ) where ζ = ξ or ζ = ξ + c τ in obtaining the estimate (3.9). And this change will not affect the result. In fact, it follows from (3.4) that 0 u + (x, t ) u + < 1. By the fact that 1 − u (ξ, t ) − φ(ξ ) = 1 − u + (x, t ), we know that 1 − u − φ > 0. When we substitute (3.9) with this change into that corresponding to (3.7), the term −u can be dropped. w (ξ +c τ ) Now we look at the last term in the right-hand side of (3.13). Note that supξ ∈R |φξ (ξ )| C 6 , w (ξ ) 1 and using the Young-inequality xy ε x2 + C (ε ) y 2 , we have
t
2μ s
φξ (ξ − c τ )u (ξ, s) + φξ (ξ )u (ξ − c τ , s − τ ) e u ξ (ξ, s) w (ξ ) dξ ds
b
0 R
t 2bε C 6
2
e2μs u ξ (s) L 2 ds + 2bC 6 C (ε ) w
0
We choose
t e 0
t
2
e2μs u (s) L 2 ds + bC 6 C (ε )
0
w
2
e2μs u 0 (s) L 2 ds. w
(3.14)
−τ
0
ε sufficiently small such that 2bε C 6 C 5 /2. It follows from (3.11) that
2μ s
0 2 2 2 u (s) L 2 ds C u 0 (0) L 2 + u 0 (s) L 2 ds . w
w
w
−τ
(3.15)
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
1103
Substituting (3.14)–(3.15) into (3.13), we obtain
e
2μt
2 C5 u ξ (t ) L 2 +
t e
2
w
2μ s
0 2 2 2 u ξ (s) L 2 ds C 7 u 0 (0) H 1 + u 0 (s) H 1 ds . w
w
w
−τ
0
Dropping the positive terms
e
2μt
t 0
e2μs u ξ (s)22 ds in the above inequality, we obtain Lw
0 2 2 2 u ξ (t ) L 2 C 7 u 0 (0) H 1 + u 0 (s) H 1 ds . w
w
(3.16)
w
−τ
Combining (3.12) with (3.16) and noting that w (ξ ) 1 on R, we obtain the following decay rate: Lemma 3.6. It holds that
0 2 2 2 2 − μ t u (t ) 1 u (t ) 1 C 8 e u 0 (0) 1 + u 0 (s) 1 ds , H H H H w
w
w
t > 0,
−τ
√
where C 8 = max{ C 1 ,
√
C 7 }.
Using Sobolev embedding theorem H 1 (R) → C 0 (R) and Lemma 3.6, we have the following stability result: Lemma 3.7. It holds that
sup u + (x, t ) − φ(x + ct ) = sup u (ξ, t ) C 9 e−μt , x∈R
ξ ∈R
t > 0,
where C 9 > 0. Step 2. Next, we prove the convergence of u − (x, t ) to φ(x + ct ). Let ξ = x + ct and
u 0 (ξ, s) = φ(x + cs) − u − 0 (x, s).
u (ξ, t ) = φ(x + ct ) − u − (x, t ),
Similar to Step 1, we have the following stability result: Lemma 3.8. It holds that
sup u − (x, t ) − φ(x + ct ) = sup u (ξ, t ) C 10 e−μt , x∈R
ξ ∈R
t > 0,
where C 10 > 0. Step 3. In the last step, we prove the convergence of u (x, t ) to φ(x + ct ). Using Lemmas 3.7 and 3.8, similar to the proof of Lemma 3.10 in [12], we can prove the convergence of u (x, t ) to
φ(x + ct ), that is
sup u (x, t ) − φ(x + ct ) C 11 e−μt , x∈R
for some C 11 > 0. The proof of Theorem 2.1 is completed.
t>0
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4. Application The above method can be used to discuss the nonlinear stability of traveling wave fronts of the following equations
ut − J ∗ u + u + dm u = εb u (x, t − τ ) ,
x ∈ R, t > 0
(4.1)
and
ut − J ∗ u + u = u (x, t ) 1 − u (x, t ) ,
x ∈ R, t > 0,
(4.2)
where functions J and b are defined as in the introduction (see (1.2) and (1.4)), constants dm and ε are positive. Eq. (4.1) can be regarded as the second version of the so-called Nicholson’s blowflies equation with diffusion which was widely studied in the literature and we can refer to Mei et al. [12], Mei and So [14], Mei, So, Li and Shen [15] and references therein. Eq. (4.2) can be regarded as the second version of the classical Logistic equation, which was studied by many authors, see Fisher [7], Kolmogorov et al. [8] and Sattinger [19]. We first consider Eq. (4.1) with the initial data
u (x, s) = u 0 (x, s),
x ∈ R, s ∈ [−τ , 0].
(4.3)
It is easily seen that Eq. (4.1) has two constant equilibria u ± , where
for b1 (u ):
u − = 0 and
u+ =
for b2 (u ):
u − = 0 and
u+ =
1 a
ln
εp
1/q
dm
ε p − dm
, 1/q
adm
,
where b1 (u ) and b2 (u ) are defined as (1.4). The existence of traveling wave fronts for Eq. (4.1) can be proved similar to Pan, Li and Lin [18]. Here we are interested in the nonlinear stability of traveling wave fronts of Eq. (4.1). Throughout this section, we assume that b (u + ) is sufficiently small and there exists a positive constant η such that
+∞
J ( y )eη y d y < 2dm − 2εb (u + ) +
1 2
.
0
Next, we consider Eq. (4.2) with the initial data
u (x, 0) = u 0 (x),
x ∈ R.
(4.4)
It is easy to see that Eq. (4.2) has two constant equilibria u − = 0 and u + = 1. One can prove the existence of traveling wave fronts for Eq. (4.2) similar to Fisher [7]. The stability of traveling wave fronts for Eq. (4.2) with J ∗ u − u replace by u was obtained by Lv and Wang [11]. In order to obtain the nonlinear stability of traveling wave fronts for Eq. (4.2), we assume +∞ that there exists a positive constant η such that 0 J ( y )eη y d y < 5/6. Now we state the stability results and the proofs are similar to that of Section 3. Theorem 4.1. Let b (u + ) be sufficiently small such that εb (u + ) < dm , and
c¯ =
1
η
2ε p − 2dm −
1 2
J ( y )eη y d y .
+ R
For any given traveling wave front φ(x + ct ) of (4.1) with speed c > max{¯c , c ∗ } and in [12], if the initial data satisfies
u − u 0 (x, s) u +
τ > 0, where c ∗ was defined as Proposition 2.1
for (x, s) ∈ R × [−τ , 0]
and the initial perturbation u 0 (x, s) − φ(x + cs) belongs to C ([−τ , 0], H 1w (R)), then the solution of (4.1) and (4.3) satisfies
u − u (x, t ) u +
for (x, t ) ∈ R × R+ ,
u (x, t ) − φ(x + ct ) ∈ C [0, +∞); H 1w (R) , where the function w (x) is defined by (2.4) with ξ0 satisfying εb (ξ0 − c τ ) < dm . Moreover, the solution u (x, t ) converges to the traveling wave front φ(x + ct ) exponentially in time:
sup u (x, t ) − φ(x + ct ) C e−μt x∈R
for some positive constants C and μ.
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
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Theorem 4.2. For any given traveling wave front φ(x + ct ) with speed
η y c > 2 max 1, J ( y )e d y , R
if the initial data satisfies
u − u 0 (x) u +
for x ∈ R
and the initial perturbation u 0 (x) − φ(x) belongs to H 1w (R), then the solution of (4.2) and (4.4) satisfies
u − u (x, t ) u +
for (x, t ) ∈ R × R+ ,
u (x, t ) − φ(x + ct ) ∈ C [0, ∞); H 1w (R) , where the function w (x) is defined by (2.4) with ξ0 satisfying φ(ξ0 ) > 2/3. Moreover, the solution u (x, t ) converges to the wave front φ(x + ct ) exponentially in time:
sup u (x, t ) − φ(x + ct ) C e−μt x∈R
for some positive constants C and μ. 5. Discussion In this paper, we have established the nonlinear stability of traveling wave fronts for nonlocal delayed reaction–diffusion equation and nonlocal KPP equation without delay. Unfortunately, we only obtained the stability of traveling wave fronts with fast speed. Just recently, Mei et al. [16] established the stability of critical wave fronts of (1.3). In our further work, we will consider the critical wave fronts of (1.1). It follows from the proof of Theorem 2.1 that the kernel function J (u ) plays an important role in establishing the stability theorem. Now we recall the restrictions about the kernel function J (u ) in the earlier results. Pan, Li and Lin [18] requested the kernel function J (u ) was an even function and satisfied the condition (H3)–(H4), that is, J (u ) J ( v ) if u , v ∈ C (R, Rn ) with u v, and R J ( y )eλ y d y < ∞ for all λ ∈ R. It is easy to see that the kernel function J (u ) must be nonnegative in paper [18]. Moreover, Bates and Chen [1,2] requested the kernel function J (u ) is a nonnegative function. Both Bates et al. [3] and Carr and Chamj [4] assumed that the kernel function J (u ) is a nonnegative even function. As in the introduction stated that there is a significant difference between the stability of traveling wave fronts for Eqs. (1.1) and (1.7). If we choose a good kernel function J (u ), we can let b − a be sufficiently small. Thus there is a significant difference between Theorem 2.1 in this paper and Theorem 5.1 in Lv and Wang [11]. Moreover, the technique in dealing with the derivative estimate in present paper can be used in paper [11] and the better result will be obtained. At last, the method we used here is also suitable to obtain the stability of traveling wave fronts for nonlocal reaction– diffusion equation with nonlocal time-delay and nonlocal delayed reaction–diffusion system, for example, nonlocal Nicholson’s blowflies equation with nonlocal time-delay (5.1) and delayed Belousov–Zhabotinskii reaction–diffusion system (5.2):
v t − J ∗ u + u + dm u = ε
b v (x − y , t − τ ) f α ( y ) d y ,
(5.1)
R y2
ε > 0, τ 0, f α ( y ) = √41π α e− 4α and ut = J ∗ u − u + u 1 − u − r v (x, t − τ ) , (x, t ) ∈ R × R+ , v t = J ∗ v − v − buv , (x, t ) ∈ R × R+
with initial data (1.6), where constants dm ,
(5.2)
with initial data
u (x, 0) = u 0 (x),
x ∈ R,
v (x, s) = v 0 (x, s),
(x, t ) ∈ R × [−τ , 0].
The existence of traveling wave fronts for Eq. (5.1) can be proved similarly to Pan, Li and Lin [18] and that for system (5.2) was proved by Pan, Li and Lin [18]. The nonlinear stability of traveling wave fronts for Nicholson’s blowflies equation with nonlocal time-delay was proved by Mei et al. [13]. In our paper [10], we studied the nonlinear stability of traveling wave fronts for delayed Belousov–Zhabotinskii reaction–diffusion system. The stability results about Eq. (5.1) and system (5.2) are similar to Theorem 2.2 in [13] and Theorem 4.2 in [10], respectively. Acknowledgments The authors are grateful to the referees for their valuable suggestions and comments on the original manuscript.
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References [1] P.W. Bates, F.X. Chen, Spectral analysis and multidimensional stability of travelling waves for nonlocal Allen–Cahn equation, J. Math. Anal. Appl. 273 (2002) 45–57. [2] P.W. Bates, F.X. Chen, Spectral analysis of traveling waves for nonlocal evolution equations, SIAM J. Math. Anal. 38 (2006) 116–126. [3] P.W. Bates, P.C. Fife, X. Ren, X. Wang, Traveling waves in a nonlocal model of phase transitions, Arch. Ration. Mech. Anal. 138 (1997) 105–136. [4] J. Carr, A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132 (2004) 2433–2439. [5] X.F. Chen, Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal evolution equations, Adv. Differential Equations 2 (1997) 125–160. [6] X.F. Chen, J.S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations 184 (2002) 549–569. [7] R.A. Fisher, The advance of advantageous genes, Ann. Eugen. 7 (1937) 355–369. [8] A.N. Kolmogorov, I.G. Petrovsky, N.S. Piskunov, Etude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Bull. Univ. Moskov. Ser. Internat. Sect. 1 (1937) 1–25. [9] G.J. Lin, Y.G. Hong, Travelling wavefronts in a vector disease model with delay, Appl. Math. Model. 32 (2008) 2831–2838. [10] G.Y. Lv, M.X. Wang, Nonlinear stability of traveling wave fronts for delayed reaction diffusion systems, submitted for publication. [11] G.Y. Lv, M.X. Wang, Nonlinear stability of travelling waves for delayed reaction diffusion equations, Nonlinearity 23 (2010) 845–873. [12] M. Mei, C.-K. Lin, C.-T. Lin, J.W.-H. So, Traveling wavefronts for time-delay reaction–diffusion equation: (I) local nonlinearity, J. Differential Equations 247 (2009) 495–510. [13] M. Mei, C.-K. Lin, C.-T. Lin, J.W.-H. So, Traveling wavefronts for time-delay reaction–diffusion equation: (II) nonlocal nonlinearity, J. Differential Equations 247 (2009) 511–529. [14] M. Mei, J.W.-H. So, Stability of strong traveling waves for a nonlocal time-delay reaction–diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 551–568. [15] M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson’s blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579–594. [16] M. Mei, C.H. Ou, X.Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction–diffusion equations, SIAM J. Math. Anal. 42 (2010) 2762–2790. [17] J.D. Murray, Mathematical Biology, Springer, Berlin/Heidelberg/New York, 1993. [18] S.X. Pan, W.T. Li, G. Lin, Traveling fronts in nonlocal delayed reaction–diffusion systems and applications, Z. Angew. Math. Phys. 60 (2009) 377–392. [19] D.H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math. 22 (1976) 312–355. [20] K.W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc. 302 (1987) 587–615. [21] H.L. Smith, X.Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction–diffusion equations, SIAM J. Math. Anal. 31 (2000) 514–534. [22] Z.C. Wang, W.T. Li, S.G. Ruan, Traveling wave fronts in reaction–diffusion systems with spatio-temporal delays, J. Differential Equations 222 (2006) 185–232. [23] Z.C. Wang, W.T. Li, S.G. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations 20 (2008) 573–607. [24] J.H. Wu, X.F. Zou, Traveling wave fronts of reaction–diffusion systems with delay, J. Dynam. Differential Equations 13 (2001) 651–687. [25] X.F. Zou, Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type, J. Comput. Appl. Math. 46 (2002) 309–321.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Nonlinear stability of traveling wave fronts for nonlocal delayed reaction–diffusion equations ✩ Guangying Lv a,∗ , Mingxin Wang b a b
College of Mathematics and Information Science, Henan University, Kaifeng 475001, PR China Science Research Center, Harbin Institute of Technology, Harbin 150080, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 28 January 2011 Available online 23 July 2011 Submitted by J. Shi
This paper is concerned with the nonlinear stability of traveling wave fronts for nonlocal delayed reaction–diffusion equation. We prove that these traveling wave fronts are exponentially stable to perturbation in some exponentially weighted L ∞ spaces, when the initial perturbation around the traveling wave fronts decays exponentially as x → −∞, but the initial perturbation can be arbitrary large in other locations. The time decay rate is also obtained by weighted energy estimates. © 2011 Elsevier Inc. All rights reserved.
Keywords: Delay Stability Nonlocal reaction–diffusion equation Traveling wave fronts Weighted energy
1. Introduction In this paper, we study nonlinear stability of traveling wave fronts for the following nonlocal delayed reaction–diffusion equation
ut = J ∗ u − u + f u (x, t ), u (x, t − τ ) ,
x ∈ R, t > 0,
τ 0, J ∈ C 1 (R) is a nonnegative even function and satisfies J ( y ) d y = 1, J ∗u= J ( x − y )u ( y , t ) d y , J ( y )eλ y d y < ∞
(1.1)
where constant
R
R
(1.2)
R
for all λ ∈ R. It is well known that there exists a unique (up to translation) wave front of Eq. (1.1) when τ = 0, see Bates et al. [3], Carr and Chamj [4], Chen [5] and references therein. The asymptotic stability of traveling wave fronts for Eq. (1.1) with τ = 0 was obtained by Chen [5] using squeezing method. Bates and Chen [1] established the multidimensional stability of traveling wave fronts for Eq. (1.1) with τ = 0 using spectral analysis. Recently, Pan, Li and Lin [18] considered Eq. (1.1) under the condition that function f satisfies quasi-monotonicity and established the existence of traveling wave fronts by using Schauder’s fixed point theorem and upper–lower solution technique. In this paper, we are interested in the nonlinear stability of traveling wave fronts for Eq. (1.1). ✩ This work was supported by PRC Grants NSFC 11071049, JSPS Innovation Program CX09B_044Z and the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1009). Corresponding author. E-mail addresses: [email protected] (G.Y. Lv), [email protected] (M.X. Wang).
*
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.07.033
©
2011 Elsevier Inc. All rights reserved.
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
1095
Traveling wave fronts of delayed reaction–diffusion equations have been studied by many authors. Schaaf [20] studied two scalar reaction–diffusion equations with a discrete delay for both Huxley nonlinearity and Fisher nonlinearity and established the existence of traveling wave fronts and uniqueness of wave speeds by a phase plane analysis method. Wu and Zou [24] considered a more general reaction–diffusion system with finite delay and obtained the existence of traveling wave fronts by using the classical monotone iteration technique with sub-supersolution method. Using the method of Wu and Zou [24], Zou [25] obtained the existence of traveling wave fronts for the delayed KPP equation, and recently, Lin and Hong [9] extends the results of [25] to a host-vector disease model
ut = du xx − au (x, t ) + bu (x, t − τ ) 1 − u (x, t ) ,
x ∈ R, t > 0,
where constants b > a 0. The study of uniqueness and asymptotic stability of traveling wave fronts become relatively more difficult. Sattinger [19] studied a reaction–diffusion system without delay. By detail spectral analysis, he proved that the traveling wave fronts were stable to perturbations in some exponentially weighted L ∞ spaces. Smith and Zhao [21] considered Eq. (1.1) with J ∗ u − u replaced by u. They first established the existence and comparison theorem of solution for a quasi-monotone reaction–diffusion bistable equation with a discrete delay and then obtained the stability of traveling wave fronts by using the elementary super-subsolution comparison and squeezing methods developed by Chen [5] (see also [6,22,23] for this technique). Just recently, Mei et al. [12] considered the so-called Nicholson’s blowflies equation with diffusion
v t − D m v xx + dm v = εb v (x, t − r ) , where constants D m > 0, dm > 0,
b1 ( v ) = p ve−av
q
(1.3)
ε > 0 and pv
or b2 ( v ) =
1 + av q
(1.4)
.
They first established a comparison principle and then proved that traveling wave fronts of Eq. (1.3) are asymptotic stable in some exponentially weighted L ∞ spaces. Lv and Wang [11] considered some more general models and established the stability of traveling wave fronts using the method developed by Mei et al. [12]. Meanwhile, Murray [17] pointed out that the general reaction–diffusion equations
ut = du xx + f (u ),
x ∈ R, t > 0
are strictly only applicable to dilute systems and population models. Moreover, time delay seems unavoidable in a real world. Thus Eq. (1.1) is a more suitable model. To our best knowledge, there is no result about the stability of traveling wave fronts for Eq. (1.1). Encouraged by papers [12,15], in this paper we study the nonlinear stability of traveling wave fronts for Eq. (1.1) with f = −au + bu (x, t − τ )(1 − u ) and a < b. The corresponding Cauchy problem is
ut = J ∗ u − u − au + bu (x, t − τ )(1 − u ), u (x, s) = u 0 (x, s),
x ∈ R, t > 0,
(1.5)
x ∈ R, s ∈ [−τ , 0].
(1.6)
It is remarked that there is some difference between the problem (1.5)–(1.6) and the following problem
ut = u xx + f u (x, t ), u (x, t − τ ) ,
x ∈ R, t > 0,
u (x, s) = u 0 (x, s),
x ∈ R, s ∈ [−τ , 0].
(1.7)
It follows from the basic estimates of paper [11] that the solution u (x, t ) of (1.7) satisfies
0 2 2 2 u ξ (s) 2 C u 0 (0) 2 + u 0 (s) 2 ds . L L L w
w
w
−τ
However, in the basic estimates of solution to the problem (1.5)–(1.6) doesn’t contain the above term, we cannot obtain the derivative estimate by the method of [11]. We shall use Young-inequality and the behavior of function u to overcome the difficulty. Throughout this paper, C > 0 denotes a generic constant, while C i (i = 1, 2, . . .) represents a specific constant. Let I be an interval, typically I = R. Denote by L 2 ( I ) the space of square integrable functions defined on I , and H k ( I ) (k 0) the Sobolev space of the L 2 -functions f (x) defined on the interval I whose derivatives Let
L 2w ( I )
2
di dxi
f (i = 1, . . . , k) also belong to L 2 ( I ).
be the weighted L -space with a weight function w (x) > 0 and its norm is defined by
f L 2w ( I ) = I
2
12
w (x) f (x) dx
.
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Let H kw ( I ) be the weighted Sobolev space with the norm given by
f Hk (I ) = w
k i =0 I
12
i
d f (x) 2
dx . w (x)
i
dx
Let T > 0 be a number and B be a Banach space. We denote by C 0 ([0, T ]; B ) the space of the B-valued continuous function on [0, T ], and by L 2 ([0, T ]; B ) the space of the B-valued L 2 -functions on [0, T ]. The corresponding spaces of the B-valued L 2 -functions on [0, ∞) are defined similarly. The rest of this paper is organized as follows. In Section 2, we introduce the traveling wave fronts and state the stability theorems. Section 3 is concerned with the proof of the stability theorem. In Section 4, the nonlinear stability of traveling wave fronts of other models with or without delay are obtained by using the similar method. This paper ends with a short discussion. 2. Preliminaries and main theorem In this section, we first recall some known results, then define weight function and last state our main theorem. Since a < b, it is easy to see that Eq. (1.5) has two nonnegative constant equilibria u − = 0 and u + = 1 − a/b. A traveling wave front of Eq. (1.5) connecting with the constant states u ± is a solution u with the form u (x, t ) = φ(x + ct ) and satisfies φ (ξ ) 0 and
c φ (ξ ) = ( J ∗ φ)(ξ ) − φ(ξ ) − aφ(ξ ) + bφ(ξ − c τ ) 1 − φ(ξ ) ,
(2.1)
φ(±∞) = u ± ,
where ξ = x + ct, = ddξ . For the special case a = 0, the existence of traveling wave front has been obtained by the authors of [18]. When the problem (2.1) has a pair of upper and lower solutions, it follows from [18] that the problem (2.1) has at least one solution. Let
¯ ) = min 1 − a/b, eλ1 (c )ξ , φ(ξ where q > 1 is large enough, equation
φ(ξ ) = max 0, eλ1 (c )ξ − qeκ λ1 (c )ξ ,
κ ∈ (1, min{2, λ2 (c )/λ1 (c )}), and λ1 (c ) and λ2 (c ) are two positive roots of the following
c (λ) = J ∗ eλ· − 1 − c λ − a + be−λc τ ,
(2.2)
and satisfy λ1 (c ) < λ2 (c ), where
J ( y )eλ y d y .
J ∗ eλ· = R
¯ ) and φ(ξ ) are a pair The existence of λ1 (c ), λ2 (c ) will be given in the following Proposition 2.1. It is easy to verify that φ(ξ of upper and lower solutions of Eq. (2.1). Proposition 2.1. Let c (λ) define as (2.2), then there exist c ∗ > 0 and λ∗ such that c (λ) = 0 has two distinct positive roots λ2 (c ) > λ∗ > λ1 (c ) for each c > c ∗ and one positive root λ∗ for c = c ∗ . And for each c c ∗ , Eq. (1.5) has a traveling wave front φ(x + ct ) satisfying φ(±∞) = u ± . Moreover, if c > c ∗ ,
lim φ(ξ )e−λ1 (c )ξ = 1,
ξ →−∞
lim φ (ξ )e−λ1 (c )ξ = λ1 (c ).
ξ →−∞
The proof of Proposition 2.1 is standard and we omit it. Throughout this paper, we always assume that there exists a positive constant
+∞ J ( y )eη y d y <
1 2
2
+ (b − a). 3
η such that (2.3)
0
We define a weight function as
w (ξ ) =
e−η(ξ −ξ0 )
for ξ ξ0 ,
1
for ξ > ξ0 ,
−a) where ξ0 is chosen to be large enough such that φ(ξ0 − c τ ) > 2(b3b . Now, we state our main theorem.
(2.4)
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
1097
Theorem 2.1. Let
c¯ =
1
2b − 2a −
η
1
∗ J ( y )eη y d y + J ∗ eλ · .
+
2
R
For any given traveling wave front φ(x + ct ) of (1.5) with speed c > max{¯c , c ∗ } and τ > 0, if the initial data satisfies
u − u 0 (x, s) u +
for (x, s) ∈ R × [−τ , 0],
and the initial perturbation u 0 (x, s) − φ(x + cs) belongs to C ([−τ , 0], H 1w (R)), then the solution of (1.5)–(1.6) satisfies
u − u (x, t ) u +
for (x, t ) ∈ R × R+ ,
u (x, t ) − φ(x + ct ) ∈ C [0, +∞); H 1w (R) , where the function w (x) is defined by (2.4). Moreover, the solution u (x, t ) converges to the traveling wave front φ(x + ct ) exponentially in time:
sup u (x, t ) − φ(x + ct ) C e−μt x∈R
for some positive constants C and μ. Remark 2.1. It is easy to show that the results of Theorem 2.1 also hold for a = 0 and b > 0. In other words, we also establish the stability of traveling wave fronts for nonlocal delayed KPP equation. Remark 2.2. It follows from the assumptions of Theorem 2.1 that know that
c η > c¯ η = 2b − 2a −
= J ∗e
λ∗ ·
> J ∗e
λ1 (c )·
1 2
+
η > λ1 (c ). In fact, from the choice of the speed c, we
∗ J ( y )eη y d y + J ∗ eλ ·
R
− 1 − a + be
−λ1 (c )c τ
− 1 − a + be
+
1 2
+b−a+b 1−e
−λ1 (c )c τ
J ( y )eη y d y
+ R
−λ1 (c )c τ
= c λ1 (c ), ∗ where we have used the facts that b > a, e−λ1 (c )c τ < 1, and J ∗ eλ · J ∗ eλ1 (c )· . In fact, direct calculations show that
J ∗e
λ·
λ
=
∞ J ( y ) ye
λy
dy =
R
J ( y ) y eλ y − e−λ y d y 0,
λ > 0.
0
Hence we have that η > λ1 (c ). We also remark that the choice of the speed c depends on we first choose the positive constant η satisfying (2.3), and then we define the speed c.
η. Actually, for any fixed b > a 0,
3. Proof of Theorem 2.1 In this section, we first establish a comparison principle for the problem (1.5)–(1.6) and then prove Theorem 2.1 by using the weighted energy method. The method we used here is similar to Mei et al. [12] and Lv and Wang [11]. Lemma 3.1. Let T > 0 and Q T = R × (0, T ]. Assume that the nonnegative function c (x, t ) is bounded for (x, t ) ∈ Q T . If the function u satisfies
ut − J ∗ u + u + c (x, t )u 0 ( 0),
(x, t ) ∈ Q T ,
u (x, 0) 0 ( 0),
x ∈ R,
then u (x, t ) 0 ( 0) for (x, t ) ∈ Q T . The proof of this lemma is easy and we omit it.
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G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
Lemma 3.2. Let
u − = 0 u 0 (x, s) u +
for (x, s) ∈ R × [−τ , 0].
Then the solution u (x, t ) of the Cauchy problem (1.5)–(1.6) satisfies
u − u (x, t ) u +
for (x, t ) ∈ R × R+ .
Proof. We first prove u (x, t ) 0. For t ∈ [0, τ ], we have u (x, t − τ ) = u 0 (x, t − τ ). Thus u (x, t ) satisfies the following equation
ut − J ∗ u + u + a + bu 0 (x, t − τ ) u = bu 0 (x, t − τ ) 0,
(x, t ) ∈ R × [0, τ ],
u (x, s) = u 0 (x, s) 0,
(x, s) ∈ R × [−τ , 0].
It follows from Lemma 3.1 that u (x, t ) 0 on R × [0, τ ]. Repeating this procedure to each of the intervals [nτ , (n + 1)τ ], n = 1, 2, . . . , we have u (x, t ) 0 on R × R+ . Next, we prove u (x, t ) u + . It is easy to see that
(u − u + )t − J ∗ (u − u + ) + u − u + + a + bu (x, t − τ ) (u − u + ) = a u (x, t − τ ) − u + , (x, t ) ∈ R × R+ , u (x, s) − u + = u 0 (x, s) − u + ,
(x, s) ∈ R × [−τ , 0].
Similar to the proof of u (x, t ) 0, we first consider t ∈ [0, τ ]. Note that 0 u 0 (x, s) u + on R × [−τ , 0], we have
⎧ ⎪ ⎨ (u − u + )t − J ∗ (u − u + ) + u − u + + a + bu 0 (x, t − τ ) (u − u + ) (x, t ) ∈ R × [0, τ ], = a u 0 (x, t − τ ) − u + 0, ⎪ ⎩ u (x, s) − u + = u 0 (x, s) − u + 0, (x, s) ∈ R × [−τ , 0].
Again by Lemma 3.1, we have u (x, t ) u + on R × [0, τ ]. Repeating the above procedure to each of the intervals [nτ , (n + 1)τ ], n = 1, 2, . . . , we have u (x, t ) u + on R × R+ . This completes the proof. 2 + Lemma 3.3 (Comparison principle). Let u − (x, t ) and u + (x, t ) be solutions of Eq. (1.5) with initial data u − 0 (x, s) and u 0 (x, s), respectively. If
+ u− u− 0 (x, s) u 0 (x, s) u +
for (x, s) ∈ R × [−τ , 0],
u − u − (x, t ) u + (x, t ) u +
for (x, t ) ∈ R × R+ .
then
Proof. It follows from Lemma 3.2 that
u − u − (x, t ) u + ,
u − u + (x, t ) u +
for (x, t ) ∈ R × R+ .
− Now we prove u − (x, t ) u + (x, t ). Let u (x, t ) = u + (x, t ) − u − (x, t ) and u 0 (x, s) = u + 0 (x, s) − u 0 (x, s), then u (x, t ) and u 0 (x, s) satisfy
ut − J ∗ u + u + a + bu + (x, t − τ ) u = bu (x, t − τ ) 1 − u − , (x, t ) ∈ R × [0, τ ], u (x, s) = u 0 (x, s),
(x, s) ∈ R × [−τ , 0].
From the assumptions, we know that u 0 (x, s) 0 on R × [−τ , 0]. Noting that u − (x, t ) u + = 1 −
− 0, (x, t ) ∈ R × [0, τ ], ut − J ∗ u + u + a + bu + 0 (x, t − τ ) u = bu 0 (x, t − τ ) 1 − u
u (x, s) = u 0 (x, s) 0,
a b
< 1, we have
(x, s) ∈ R × [−τ , 0].
Hence by Lemma 3.1, we have u (x, t ) 0 on R × [0, τ ], that is, u − (x, t ) u + (x, t ) on R × [0, τ ]. Repeating the above procedure to each of the intervals [nτ , (n + 1)τ ], n = 1, 2, . . . , we have u − (x, t ) u + (x, t ) on R × R+ . This completes the proof.
2
Following the idea of [12,15], we shall use the comparison principle and weighted energy method to prove Theorem 2.1. Let the initial data u 0 (x, s) satisfies
u − = 0 u 0 (x, s) u + Define
for (x, s) ∈ R × [−τ , 0].
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
1099
u− 0 (x, s) = min u 0 (x, s), φ(x + cs) ,
(x, s) ∈ R × [−τ , 0], u 0 (x, s) = max u 0 (x, s), φ(x + cs) , (x, s) ∈ R × [−τ , 0].
+
Obviously, + u− u− 0 (x, s) u 0 (x, s) u 0 (x, s) u + , −
(x, s) ∈ R × [−τ , 0],
+
u − u 0 (x, s) φ(x + cs) u 0 (x, s) u + ,
(3.1)
(x, s) ∈ R × [−τ , 0].
(3.2)
+ Let u − (x, t ) and u + (x, t ) be the solutions of Eqs. (1.5)–(1.6) with initial data u − 0 (x, s) and u 0 (x, s), respectively. It follows
comparison principle that
u − u − (x, t ) u (x, t ) u + (x, t ) u + ,
(x, s) ∈ R × R+ ,
u − u − (x, t ) φ(x + ct ) u + (x, t ) u + ,
(3.3)
(x, s) ∈ R × R+ .
(3.4)
Proof of Theorem 2.1. We prove Theorem 2.1 in three steps. Step 1. We first prove the convergence of u + (x, t ) to φ(x + ct ). Let ξ = x + ct and
u 0 (ξ, s) = u + 0 (x, s) − φ(x + cs).
u (ξ, t ) = u + (x, t ) − φ(x + ct ), Then by (3.2) and (3.4), we have
u (ξ, t ) 0,
u 0 (ξ, s) 0.
Moreover, u (ξ, t ) satisfies
ut + cu ξ − J ∗ u + u + a + bφ(ξ − c τ ) u = bu (ξ − c τ , t − τ )(1 − u − φ),
(ξ, t ) ∈ R × R+ ,
u (ξ, s) = u 0 (ξ, s),
(ξ, s) ∈ R × [−τ , 0].
Let w (ξ ) > 0 be the weight function defined in (2.4). Multiplying (3.5) by e2μt w (ξ )u (ξ, t ) where later in Lemma 3.5, we have
1 2
e2μt wu 2
+ t
c 2
e2μt wu 2
(3.5)
μ > 0 will be specified
− e2μt w (ξ )u (ξ, t ) ξ
J ( y )u (ξ − y , t ) d y R
c w + − − μ + 1 + a + bφ(ξ − c τ ) wu 2 e2μt 2 w
= b(1 − φ) wuu (ξ − c τ , t − τ )e2μt − bu (ξ − c τ , t − τ )u 2 we2μt .
(3.6)
Integrating (3.6) with respect to (ξ, t ) over R × [0, t ] and dropping the negative term
t bu (ξ − c τ , s − τ )u 2 (ξ, s) w (ξ )e2μs dξ ds,
− 0 R
we obtain
2
e2μt u (t ) L 2 − 2
t e2μs w (ξ )u (ξ, s)
w
0 R
t +
J ( y )u (ξ − y , s) d y dξ ds R
w e2μs −c − 2μ + 2 + 2a + 2bφ(ξ − c τ ) w (ξ )u 2 (ξ, s) dξ ds w
0 R
2 u 0 (0) L 2 + 2b
t
e2μs 1 − φ(ξ ) w (ξ )u (ξ, s)u (ξ − c τ , s − τ ) dξ ds.
w
0 R
By using the Cauchy–Schwarz inequality 2xy x2 + y 2 , we can estimate that
(3.7)
1100
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
t e2μs w (ξ )u (ξ, s)
2 0 R
J ( y )u (ξ − y , s) d y dξ ds R
t
t e2μs w (ξ )u 2 (ξ, s) dξ ds +
e2μs w (ξ )
0 R
0 R
t
t e2μs w (ξ )u 2 (ξ, s) dξ ds +
= 0 R
J ( y )u 2 (ξ − y , s) d y dξ ds R
e2μs w (ξ )u 2 (ξ, s)
0 R
J ( y)
w (ξ + y )
R
w (ξ )
d y dξ ds
(3.8)
and
t
e2μs 1 − φ(ξ ) w (ξ )u (ξ, s)u (ξ − c τ , s − τ ) dξ ds
2b 0 R
t
e2μs 1 − φ(ξ ) w (ξ ) u 2 (ξ, s) + u 2 (ξ − c τ , s − τ ) dξ ds
b 0 R
t =b
e
2μ s
1 − φ(ξ ) w (ξ )u 2 (ξ, s) dξ ds + be2μτ
e2μs 1 − φ(ξ + c τ ) w (ξ + c τ )u 2 (ξ, s) dξ ds −τ R
0 R
t b
t −τ
e
2μ s
1 − φ(ξ ) w (ξ )u 2 (ξ, s) dξ ds + be2μτ
0 R
t
e2μs 1 − φ(ξ + c τ ) w (ξ + c τ )u 2 (ξ, s) dξ ds 0 R
0
+ be2μτ
e2μs 1 − φ(ξ + c τ ) w (ξ + c τ )u 20 (ξ, s) dξ ds.
(3.9)
−τ R
Submitting (3.8) and (3.9) into (3.7) and using 0 φ(ξ ) 1 and
e
2μt
2 u (t ) L 2 +
μ ∈ (0, μ1 ) (see Lemma 3.4), we have
t e2μs B μ, w (ξ ) w (ξ )u 2 (ξ, s) dξ ds
w
0 R
0
2 u 0 (0) L 2 + be2μτ
−τ R
C1
e2μs 1 − φ(ξ + c τ ) w (ξ + c τ )u 2 (ξ, s) dξ ds
w
u 0 (0)22 + L
0
w
u 0 (s)22 ds , L
(3.10)
w
−τ
where
w (ξ + c τ ) , w (ξ ) w (ξ + y ) w (ξ ) + 1 + 2a + 2bφ(ξ − c τ ) − A w (ξ ) = −c J ( y) dy w (ξ ) w (ξ )
B μ, w (ξ ) = A w (ξ ) − 2μ − b e2μτ − 1 1 − φ(ξ + c τ )
R
w (ξ + c τ ) . − b 1 − φ(ξ ) − b 1 − φ(ξ + c τ ) w (ξ ) In order to get the basic estimate, we must prove B μ, w (ξ ) C > 0 for some constant C . For this aim we need the following key lemma. Lemma 3.4. Let c > c¯ , then there exists positive constant C 2 such that
A w (ξ ) C 2 ,
ξ ∈ R.
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
Proof. Case 1: ξ ξ0 . It follows from (2.4) that
A w (ξ ) = −c
w (ξ ) w (ξ )
w (ξ +c τ ) w (ξ )
1. Noting that 0 φ(ξ ) 1 − a/b and J (− y ) = J ( y ), we have
+ 1 + 2a + 2bφ(ξ − c τ ) −
J ( y)
w (ξ + y ) w (ξ )
R
dy
w (ξ + c τ ) − b 1 − φ(ξ ) − b 1 − φ(ξ + c τ ) w (ξ )
= c η + 1 + 2a + 2bφ(ξ − c τ ) − eη(ξ −ξ0 ) −
ξ0 −ξ
J ( y ) e−η y − eη(ξ −ξ0 ) d y
−∞
w (ξ + c τ ) − b 1 − φ(ξ ) − b 1 − φ(ξ + c τ ) w (ξ ) 1 J ( y )eη y d y − 2b c η + + 2a − 2
R
:= C 3 > 0, where we have used c > c¯ . Case 2: ξ > ξ0 . In this case, w (ξ ) = w (ξ + c τ ) = 1. Since φ(ξ ) is increasing on R and J (− y ) = J ( y ), we obtain ξ0 −ξ
A w (ξ ) = 2a + 2bφ(ξ − c τ ) −
J ( y ) e−η y − 1 d y − b 1 − φ(ξ ) − b 1 − φ(ξ + c τ )
−∞
0 2bφ(ξ0 − c τ ) − b + a −
J ( y ) e− η y − 1 d y
−∞
0 = 2bφ(ξ0 − c τ ) − b + a −
J ( y )e−η y d y +
−∞
1 2
:= C 4 > 0 by the condition (2.3). Finally, let C 2 := min{C 3 , C 4 }. Then we have A η, w (ξ ) C 2 > 0.
2
Lemma 3.5. Let μ1 > 0 be the unique root of the following equation
C 2 /2 − 2μ1 − b e2μ1 τ − 1 = 0. Then B μ, w (ξ ) C 5 := C 2 /2 > 0 on R, for 0 < μ μ1 . Proof. Note that 0 φ(ξ ) 1 on R and
w (ξ + c τ ) w (ξ )
⎧ −η c τ < 1 ⎪ for ξ ξ0 − c τ , ⎨e = eη(ξ −ξ0 ) 1 for ξ0 − c τ < ξ ξ0 , ⎪ ⎩ 1 for ξ > ξ0 .
It is easy to see that, for 0 < μ μ1 ,
w (ξ + c τ ) w (ξ )
B η,μ, w (ξ ) = A η, w (ξ ) − 2μ − b e2μτ − 1 1 − φ(ξ + c τ )
C 2 /2 − 2μ − b e2μτ − 1 := C 5 > 0. The proof is completed.
2
1101
1102
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
Submitting this result into (3.10), we have
e
2μt
2 u (t ) L 2 + C 5
t e
w
2μ s
0 2 2 2 u (s) L 2 ds C 1 u 0 (0) L 2 + u 0 (s) L 2 ds . w
w
−τ
0
t
Dropping the positive terms
e
2μt
(3.11)
w
0
e2μs u (s)22 ds, we obtain the basic estimate Lw
0 2 2 2 u (t ) L 2 C 1 u 0 (0) L 2 + u 0 (s) L 2 ds . w
w
(3.12)
w
−τ
Next, we differentiate (3.5) with respect to ξ , and then multiply the resulting equation by e2μt w (ξ )u ξ (ξ, t ) where w (ξ ) again denotes the function in (2.4), with the same value for ξ0 , we obtain
1 2
e2μt wu 2ξ
+ t
+ −c
w 2w
c 2
e2μt wu 2ξ
− e2μt w (ξ )u ξ (ξ, t ) ξ
J ( y )u ξ (ξ − y , t ) d y R
− μ + 1 + a + bφ(ξ − c τ ) wu 2ξ e2μt
= b(1 − φ − u ) wu ξ u ξ (ξ − c τ , t − τ )e2μt − bu (ξ − c τ , t − τ )u 2ξ we2μt − bφξ (ξ − c τ )uu ξ we2μt − bφξ u ξ we2μt u (ξ − c τ , t − τ ). Integrating over R × [0, t ] and carrying out similar steps to those that led to (3.11), we obtain
e
2μt
2 u ξ (t ) L 2 + C 5
t
2
e2μs u ξ (s) L 2 ds
w
w
0
u 0ξ (0)22 + L
C1
0
w
u 0ξ (s)22 ds L
w
−τ
t −b
φξ (ξ − c τ )u (ξ, s) + φξ (ξ )u (ξ − c τ , s − τ ) e2μs u ξ (ξ, s) w (ξ ) dξ ds.
(3.13)
0 R
It is remarked that there is a little difference in getting (3.11) and (3.13). It is easy to see that 1 − φ(ζ ) will be replaced by 1 − u (ζ, t ) − φ(ζ ) where ζ = ξ or ζ = ξ + c τ in obtaining the estimate (3.9). And this change will not affect the result. In fact, it follows from (3.4) that 0 u + (x, t ) u + < 1. By the fact that 1 − u (ξ, t ) − φ(ξ ) = 1 − u + (x, t ), we know that 1 − u − φ > 0. When we substitute (3.9) with this change into that corresponding to (3.7), the term −u can be dropped. w (ξ +c τ ) Now we look at the last term in the right-hand side of (3.13). Note that supξ ∈R |φξ (ξ )| C 6 , w (ξ ) 1 and using the Young-inequality xy ε x2 + C (ε ) y 2 , we have
t
2μ s
φξ (ξ − c τ )u (ξ, s) + φξ (ξ )u (ξ − c τ , s − τ ) e u ξ (ξ, s) w (ξ ) dξ ds
b
0 R
t 2bε C 6
2
e2μs u ξ (s) L 2 ds + 2bC 6 C (ε ) w
0
We choose
t e 0
t
2
e2μs u (s) L 2 ds + bC 6 C (ε )
0
w
2
e2μs u 0 (s) L 2 ds. w
(3.14)
−τ
0
ε sufficiently small such that 2bε C 6 C 5 /2. It follows from (3.11) that
2μ s
0 2 2 2 u (s) L 2 ds C u 0 (0) L 2 + u 0 (s) L 2 ds . w
w
w
−τ
(3.15)
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
1103
Substituting (3.14)–(3.15) into (3.13), we obtain
e
2μt
2 C5 u ξ (t ) L 2 +
t e
2
w
2μ s
0 2 2 2 u ξ (s) L 2 ds C 7 u 0 (0) H 1 + u 0 (s) H 1 ds . w
w
w
−τ
0
Dropping the positive terms
e
2μt
t 0
e2μs u ξ (s)22 ds in the above inequality, we obtain Lw
0 2 2 2 u ξ (t ) L 2 C 7 u 0 (0) H 1 + u 0 (s) H 1 ds . w
w
(3.16)
w
−τ
Combining (3.12) with (3.16) and noting that w (ξ ) 1 on R, we obtain the following decay rate: Lemma 3.6. It holds that
0 2 2 2 2 − μ t u (t ) 1 u (t ) 1 C 8 e u 0 (0) 1 + u 0 (s) 1 ds , H H H H w
w
w
t > 0,
−τ
√
where C 8 = max{ C 1 ,
√
C 7 }.
Using Sobolev embedding theorem H 1 (R) → C 0 (R) and Lemma 3.6, we have the following stability result: Lemma 3.7. It holds that
sup u + (x, t ) − φ(x + ct ) = sup u (ξ, t ) C 9 e−μt , x∈R
ξ ∈R
t > 0,
where C 9 > 0. Step 2. Next, we prove the convergence of u − (x, t ) to φ(x + ct ). Let ξ = x + ct and
u 0 (ξ, s) = φ(x + cs) − u − 0 (x, s).
u (ξ, t ) = φ(x + ct ) − u − (x, t ),
Similar to Step 1, we have the following stability result: Lemma 3.8. It holds that
sup u − (x, t ) − φ(x + ct ) = sup u (ξ, t ) C 10 e−μt , x∈R
ξ ∈R
t > 0,
where C 10 > 0. Step 3. In the last step, we prove the convergence of u (x, t ) to φ(x + ct ). Using Lemmas 3.7 and 3.8, similar to the proof of Lemma 3.10 in [12], we can prove the convergence of u (x, t ) to
φ(x + ct ), that is
sup u (x, t ) − φ(x + ct ) C 11 e−μt , x∈R
for some C 11 > 0. The proof of Theorem 2.1 is completed.
t>0
2
1104
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
4. Application The above method can be used to discuss the nonlinear stability of traveling wave fronts of the following equations
ut − J ∗ u + u + dm u = εb u (x, t − τ ) ,
x ∈ R, t > 0
(4.1)
and
ut − J ∗ u + u = u (x, t ) 1 − u (x, t ) ,
x ∈ R, t > 0,
(4.2)
where functions J and b are defined as in the introduction (see (1.2) and (1.4)), constants dm and ε are positive. Eq. (4.1) can be regarded as the second version of the so-called Nicholson’s blowflies equation with diffusion which was widely studied in the literature and we can refer to Mei et al. [12], Mei and So [14], Mei, So, Li and Shen [15] and references therein. Eq. (4.2) can be regarded as the second version of the classical Logistic equation, which was studied by many authors, see Fisher [7], Kolmogorov et al. [8] and Sattinger [19]. We first consider Eq. (4.1) with the initial data
u (x, s) = u 0 (x, s),
x ∈ R, s ∈ [−τ , 0].
(4.3)
It is easily seen that Eq. (4.1) has two constant equilibria u ± , where
for b1 (u ):
u − = 0 and
u+ =
for b2 (u ):
u − = 0 and
u+ =
1 a
ln
εp
1/q
dm
ε p − dm
, 1/q
adm
,
where b1 (u ) and b2 (u ) are defined as (1.4). The existence of traveling wave fronts for Eq. (4.1) can be proved similar to Pan, Li and Lin [18]. Here we are interested in the nonlinear stability of traveling wave fronts of Eq. (4.1). Throughout this section, we assume that b (u + ) is sufficiently small and there exists a positive constant η such that
+∞
J ( y )eη y d y < 2dm − 2εb (u + ) +
1 2
.
0
Next, we consider Eq. (4.2) with the initial data
u (x, 0) = u 0 (x),
x ∈ R.
(4.4)
It is easy to see that Eq. (4.2) has two constant equilibria u − = 0 and u + = 1. One can prove the existence of traveling wave fronts for Eq. (4.2) similar to Fisher [7]. The stability of traveling wave fronts for Eq. (4.2) with J ∗ u − u replace by u was obtained by Lv and Wang [11]. In order to obtain the nonlinear stability of traveling wave fronts for Eq. (4.2), we assume +∞ that there exists a positive constant η such that 0 J ( y )eη y d y < 5/6. Now we state the stability results and the proofs are similar to that of Section 3. Theorem 4.1. Let b (u + ) be sufficiently small such that εb (u + ) < dm , and
c¯ =
1
η
2ε p − 2dm −
1 2
J ( y )eη y d y .
+ R
For any given traveling wave front φ(x + ct ) of (4.1) with speed c > max{¯c , c ∗ } and in [12], if the initial data satisfies
u − u 0 (x, s) u +
τ > 0, where c ∗ was defined as Proposition 2.1
for (x, s) ∈ R × [−τ , 0]
and the initial perturbation u 0 (x, s) − φ(x + cs) belongs to C ([−τ , 0], H 1w (R)), then the solution of (4.1) and (4.3) satisfies
u − u (x, t ) u +
for (x, t ) ∈ R × R+ ,
u (x, t ) − φ(x + ct ) ∈ C [0, +∞); H 1w (R) , where the function w (x) is defined by (2.4) with ξ0 satisfying εb (ξ0 − c τ ) < dm . Moreover, the solution u (x, t ) converges to the traveling wave front φ(x + ct ) exponentially in time:
sup u (x, t ) − φ(x + ct ) C e−μt x∈R
for some positive constants C and μ.
G.Y. Lv, M.X. Wang / J. Math. Anal. Appl. 385 (2012) 1094–1106
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Theorem 4.2. For any given traveling wave front φ(x + ct ) with speed
η y c > 2 max 1, J ( y )e d y , R
if the initial data satisfies
u − u 0 (x) u +
for x ∈ R
and the initial perturbation u 0 (x) − φ(x) belongs to H 1w (R), then the solution of (4.2) and (4.4) satisfies
u − u (x, t ) u +
for (x, t ) ∈ R × R+ ,
u (x, t ) − φ(x + ct ) ∈ C [0, ∞); H 1w (R) , where the function w (x) is defined by (2.4) with ξ0 satisfying φ(ξ0 ) > 2/3. Moreover, the solution u (x, t ) converges to the wave front φ(x + ct ) exponentially in time:
sup u (x, t ) − φ(x + ct ) C e−μt x∈R
for some positive constants C and μ. 5. Discussion In this paper, we have established the nonlinear stability of traveling wave fronts for nonlocal delayed reaction–diffusion equation and nonlocal KPP equation without delay. Unfortunately, we only obtained the stability of traveling wave fronts with fast speed. Just recently, Mei et al. [16] established the stability of critical wave fronts of (1.3). In our further work, we will consider the critical wave fronts of (1.1). It follows from the proof of Theorem 2.1 that the kernel function J (u ) plays an important role in establishing the stability theorem. Now we recall the restrictions about the kernel function J (u ) in the earlier results. Pan, Li and Lin [18] requested the kernel function J (u ) was an even function and satisfied the condition (H3)–(H4), that is, J (u ) J ( v ) if u , v ∈ C (R, Rn ) with u v, and R J ( y )eλ y d y < ∞ for all λ ∈ R. It is easy to see that the kernel function J (u ) must be nonnegative in paper [18]. Moreover, Bates and Chen [1,2] requested the kernel function J (u ) is a nonnegative function. Both Bates et al. [3] and Carr and Chamj [4] assumed that the kernel function J (u ) is a nonnegative even function. As in the introduction stated that there is a significant difference between the stability of traveling wave fronts for Eqs. (1.1) and (1.7). If we choose a good kernel function J (u ), we can let b − a be sufficiently small. Thus there is a significant difference between Theorem 2.1 in this paper and Theorem 5.1 in Lv and Wang [11]. Moreover, the technique in dealing with the derivative estimate in present paper can be used in paper [11] and the better result will be obtained. At last, the method we used here is also suitable to obtain the stability of traveling wave fronts for nonlocal reaction– diffusion equation with nonlocal time-delay and nonlocal delayed reaction–diffusion system, for example, nonlocal Nicholson’s blowflies equation with nonlocal time-delay (5.1) and delayed Belousov–Zhabotinskii reaction–diffusion system (5.2):
v t − J ∗ u + u + dm u = ε
b v (x − y , t − τ ) f α ( y ) d y ,
(5.1)
R y2
ε > 0, τ 0, f α ( y ) = √41π α e− 4α and ut = J ∗ u − u + u 1 − u − r v (x, t − τ ) , (x, t ) ∈ R × R+ , v t = J ∗ v − v − buv , (x, t ) ∈ R × R+
with initial data (1.6), where constants dm ,
(5.2)
with initial data
u (x, 0) = u 0 (x),
x ∈ R,
v (x, s) = v 0 (x, s),
(x, t ) ∈ R × [−τ , 0].
The existence of traveling wave fronts for Eq. (5.1) can be proved similarly to Pan, Li and Lin [18] and that for system (5.2) was proved by Pan, Li and Lin [18]. The nonlinear stability of traveling wave fronts for Nicholson’s blowflies equation with nonlocal time-delay was proved by Mei et al. [13]. In our paper [10], we studied the nonlinear stability of traveling wave fronts for delayed Belousov–Zhabotinskii reaction–diffusion system. The stability results about Eq. (5.1) and system (5.2) are similar to Theorem 2.2 in [13] and Theorem 4.2 in [10], respectively. Acknowledgments The authors are grateful to the referees for their valuable suggestions and comments on the original manuscript.
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References [1] P.W. Bates, F.X. Chen, Spectral analysis and multidimensional stability of travelling waves for nonlocal Allen–Cahn equation, J. Math. Anal. Appl. 273 (2002) 45–57. [2] P.W. Bates, F.X. Chen, Spectral analysis of traveling waves for nonlocal evolution equations, SIAM J. Math. Anal. 38 (2006) 116–126. [3] P.W. Bates, P.C. Fife, X. Ren, X. Wang, Traveling waves in a nonlocal model of phase transitions, Arch. Ration. Mech. Anal. 138 (1997) 105–136. [4] J. Carr, A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132 (2004) 2433–2439. [5] X.F. Chen, Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal evolution equations, Adv. Differential Equations 2 (1997) 125–160. [6] X.F. Chen, J.S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations 184 (2002) 549–569. [7] R.A. Fisher, The advance of advantageous genes, Ann. Eugen. 7 (1937) 355–369. [8] A.N. Kolmogorov, I.G. Petrovsky, N.S. Piskunov, Etude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Bull. Univ. Moskov. Ser. Internat. Sect. 1 (1937) 1–25. [9] G.J. Lin, Y.G. Hong, Travelling wavefronts in a vector disease model with delay, Appl. Math. Model. 32 (2008) 2831–2838. [10] G.Y. Lv, M.X. Wang, Nonlinear stability of traveling wave fronts for delayed reaction diffusion systems, submitted for publication. [11] G.Y. Lv, M.X. Wang, Nonlinear stability of travelling waves for delayed reaction diffusion equations, Nonlinearity 23 (2010) 845–873. [12] M. Mei, C.-K. Lin, C.-T. Lin, J.W.-H. So, Traveling wavefronts for time-delay reaction–diffusion equation: (I) local nonlinearity, J. Differential Equations 247 (2009) 495–510. [13] M. Mei, C.-K. Lin, C.-T. Lin, J.W.-H. So, Traveling wavefronts for time-delay reaction–diffusion equation: (II) nonlocal nonlinearity, J. Differential Equations 247 (2009) 511–529. [14] M. Mei, J.W.-H. So, Stability of strong traveling waves for a nonlocal time-delay reaction–diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 551–568. [15] M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson’s blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579–594. [16] M. Mei, C.H. Ou, X.Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction–diffusion equations, SIAM J. Math. Anal. 42 (2010) 2762–2790. [17] J.D. Murray, Mathematical Biology, Springer, Berlin/Heidelberg/New York, 1993. [18] S.X. Pan, W.T. Li, G. Lin, Traveling fronts in nonlocal delayed reaction–diffusion systems and applications, Z. Angew. Math. Phys. 60 (2009) 377–392. [19] D.H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math. 22 (1976) 312–355. [20] K.W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc. 302 (1987) 587–615. [21] H.L. Smith, X.Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction–diffusion equations, SIAM J. Math. Anal. 31 (2000) 514–534. [22] Z.C. Wang, W.T. Li, S.G. Ruan, Traveling wave fronts in reaction–diffusion systems with spatio-temporal delays, J. Differential Equations 222 (2006) 185–232. [23] Z.C. Wang, W.T. Li, S.G. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations 20 (2008) 573–607. [24] J.H. Wu, X.F. Zou, Traveling wave fronts of reaction–diffusion systems with delay, J. Dynam. Differential Equations 13 (2001) 651–687. [25] X.F. Zou, Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type, J. Comput. Appl. Math. 46 (2002) 309–321.