Stability of traveling waves in a population dynamics model with spatio-temporal delay

Linear Algebra Applications Nonlinear Analysis and 132 its (2016) 183–195 466 (2015) 102–116

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Linear Algebra and its Applications Nonlinear Analysis www.elsevier.com/locate/laa www.elsevier.com/locate/na

Inverse waves eigenvalue problem of dynamics Jacobi matrix Stability of traveling in a population model with with mixed data spatio-temporal delay Yun-Rui Yang ∗ , LiYing Liu Wei 1 School of Mathematics Department and Physics,ofLanzhou Jiaotong University, Lanzhou, Gansu 730070, PR China Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China

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Article history: In this paper, we are concerned with the stability of all traveling waves for a poputhisspatio-temporal paper, the inverse eigenvalue of reconstructing Received 31 March 2015Article history: lation dynamics modelInwith delay. By theproblem weighted-energy method Received 16 January 2014 Accepted 2 November 2015 Jacobi matrix from exponential its eigenvalues, its leading principal combining comparisona principle, the global stability of all traveling Accepted 20 September 2014 Communicated by Enzo Mitidieri submatrix andeven partincluding of the the eigenvalues of itswhose submatrix waves for the model is established, slower waves wave Available online 22 October 2014 is considered. The necessary and sufficient conditions for speed is close to the critical speed. Submitted by Y. Wei MSC: the existence and uniqueness of the solution are reserved. derived. © 2015 Elsevier Ltd. All rights 34K18 Furthermore, a numerical algorithm and some numerical MSC: 37G10 examples are given. 15A18 37G05 © 2014 Published by Elsevier Inc. 15A57 Keywords: Traveling waves Keywords: Stability Jacobi matrix Comparison principle Eigenvalue Weighted-energy method Inverse problem Submatrix

1. Introduction As is well known, reaction–diffusion equations are often used to model and describe the practical problems we meet in ecology, biology, population dynamics, chemistry, and so on. For example, based on the experimental data of Nicholson for the population of the Australian sheep-blowfly, Gurney et al. [4] originally established the following population dynamics model without diffusion, which is the so-called Nicholson’s blowflies model du(t) = −δu(t) + pu(t − τ )e−au(t−τ ) , dt

1

E-mail address: [email protected].

Tel.:rate +86of 13914485239. where δ > 0 is the death the mature population, τ > 0 is the maturation delay, the time required for a newborn to become matured, and a > 0 is a constant. p > 0 is the impact of the death on the immature http://dx.doi.org/10.1016/j.laa.2014.09.031 ) Published by Elsevier Inc. 2014 population and u(t 0024-3795/© − τ )e−au(t−τ is Nicholson’s birth function. Considering spatial variability of blowflies,

∗ Corresponding author. E-mail address: [email protected] (Y.-R. Yang).

http://dx.doi.org/10.1016/j.na.2015.11.006 0362-546X/© 2015 Elsevier Ltd. All rights reserved.

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many problems are investigated about the diffusive Nicholson’s blowflies model ∂u(t, x) ∂ 2 u(t, x) − δu(t, x) + pu(t − τ, x)e−au(t−τ,x) , = ∂t ∂x2 see [21–23]. However, more natural attention is paid to the modeling of the time delays to incorporate associated non-local spatial terms which account for the drift of blowflies to their present position from their possible positions at previous time, i.e., the following Nicholson’s blowflies model with spatio-temporal delay ∂ 2 u(t, x) ∂u(t, x) − δu(t, x) + p(g ∗ u)(t, x)e−a(g∗u)(t,x) . = ∂t ∂x2 The convolution g ∗ u is denoted by  t  +∞ g(t − s, x − y)u(s, y)dyds, (g ∗ u)(t, x) = −∞

(1.1)

−∞

and g(t, x) denote different kinds of spatio-temporal delay kernels which are usually seen in some literature t t t [9,10,24]. For example, when (g ∗ u)(t, x) = −∞ g(t − s)u(s, x)ds and g(t) = τ1 e− τ or g(t) = τt2 e− τ , Gourley [3] proved the existence of traveling wave fronts for Eq. (1.1) with those kernel functions using linear chain trick and geometric singular perturbation theory after there is a way to recast (1.1) into a non-delay finite dimensional ordinary differential system. Unfortunately, sometimes there is no way to recast (1.1) with special kernel functions into a non-delay finite dimensional ordinary differential system, such as g(t, x) = δ(t − τ ) √

x2 1 e− 4ρ , 4πρ

τ > 0, ρ > 0.

(1.2)

Thus the methods of linear chain trick and geometric singular perturbation theory have no effect. Recently, Lin [10] employs the monotone iteration technique as well as the upper and lower solution method developed by Wang et al. [24] to obtain the existence of traveling wave solutions for (1.1) with the kernel functions (1.2). In this paper, we continue to study (1.1) with the kernel function (1.2) and consider the Cauchy problem to (1.1) under initial conditions u(s, x) = u0 (s, x),

s ∈ [−τ, 0], x ∈ R.

(1.3)

Namely, we provide a stability analysis of traveling wave solutions to (1.1). For reaction–diffusion equations with delay, to our knowledge, there are few results on the stability of traveling waves, see [8,12–17,19,20]. Schaaf [19] first established the local stability of traveling wave solutions by the spectrum analysis method for Fisher–Kpp nonlinearity and the equilibrium u− = 0 is a stable node. Later, Smith and Zhao [20] obtained a globally exponential stability result for bistable nonlinearity by the “squeezing technique” in [1]. But the above methods cannot be applied to the nonlinearity in (1.1), because here u− = 0 is an unstable node. Fortunately, this problem can be solved by introducing a proper weighted function [18] and adopting the weighted energy method developed by Mei [8,12–17]. Eq. (1.1) with the kernel function g(t, x) = δ(x)δ(t − τ )

(1.4)

has been extensively studied recently, see [12–17] and the references therein. By using the weighted energy method, Mei et al. [17] proved that the wavefronts of (1.1) with (1.4) are stable for the large wave speed (i.e., c > c∗ , c∗ is the critical wave speed) and small initial perturbation. But for the small wave speed (i.e., c is close to c∗ ) and large initial perturbation (i.e., the initial perturbation around the wavefront decays to zero exponentially in space as x → −∞, but it can be allowed arbitrary large in other locations), the results (1.1) with (1.4) in [12–14] develop and improve the previous results in [16,17] by the help of new approach developed in [12] which is a combination of comparison principle and the technical weighted-energy method. Especially in Mei’s [15], for nonlocal Nicholson’s nonlinearity different from ours in this paper, Mei established the stability for all waves even including the slower waves whose wave speed is close to the critical

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speed as long as selecting an ideal weight function. Motivated by this idea, here we study the stability of traveling wavefronts of (1.1) with (1.2) for the large initial perturbation and the wave speed c > c∗ including those slower waves by introducing a non-piecewise weight function firstly, which is exponentially decayed and related to the critical wave speed c∗ . Secondly, combining the weight function with some new techniques, the difficulty of the energy estimates caused by the nonlocal integral term can be overcome, and the convergence result of solutions is obtained on any interval I = (−∞, ξ∗ ] for some large ξ∗ ≫ 1. Next, we extend the L∞ -convergence result to the whole space R by an integral technique or Halanay’s inequality. Finally, by Comparison principle and the squeeze theorem, we obtain the stability for all waves of (1.1) with (1.2). Notations: Throughout this paper, C > 0 denotes a generic constant, Ci > 0(i = 1, 2, . . .) represents a specific constant. Let I be an interval. L2 (I) is the space of the square integrable functions defined on I, and H k (I)(k ≥ 0) is the Sobolev space of the L2 -functions f (x) defined on the interval I whose derivatives di 2 2 2 dxi f (i = 1, 2, . . . , k) also belong to L (I). Lw (I) denotes the weighted L -space with a weight function w(x) > 0 and its norm is defined by   12 2 , ∥f ∥L2w = w(x)|f (x)| dx I

Hwk (I)

is the weighted Sobolev space with the norm given by  k   i 2  21   d  ∥f ∥Hwk = w(x)  i f (x) dx . dx i=0 I

Let T > 0 be a number and B be a Banach space. We denote by C([0, T ]; B) the space of the B-valued continuous functions on [0, T ].L2 ([0, T ]; B) as the space of the B-valued L2 -functions on [0, T ]. The corresponding spaces of B-valued functions on [0, ∞) are defined similarly. This paper is organized as follows. In Section 2, we introduce some preliminaries and state our stability result. In Section 3, we prove our main result on the globally exponential stability of monotone traveling wave solutions after establishing the boundedness of solutions and comparison principle. 2. Preliminaries and main result Notice that Eq. (1.1) has two constant equilibria u± , where 1 p ln . (2.1) a δ If p > δ, then u+ > u− . A traveling wavefront of Eq. (1.1) connecting with u− and u+ is a monotone solution u(t, x) = φ(x + ct) satisfying the following ordinary differential equation  cφ′ (ξ) − φ′′ (ξ) + δφ(ξ) = p[(g ∗ φ)(ξ)]e−a[(g∗φ)(ξ)] , (2.2) φ(±∞) = u± , u− = 0

and u+ =

 +∞ 1 − y2 where ξ = x + ct, (g ∗ φ)(ξ) = −∞ √4πρ e 4ρ φ(ξ − cτ − y)dy. It is not difficult to examine that  +∞ 1 − y2 √ e 4ρ dy = 1. −∞ 4πρ By using the monotone iteration technique as well as the upper and lower solution method (see the early work by Wang et al. [24]), Lin [10] proved the existence of the traveling wavefronts for (1.1) with the kernel function (1.2). Proposition 2.1 (Existence of Traveling Wavefronts). Assume that 1<

p ≤ e. δ

(2.3)

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Then there exist c∗ > 0 and a corresponding number 0 < λ1 (c) < λ∗ = λ∗ (c∗ ) < λ2 (c) satisfying ∂ ∆(λ∗ , c∗ ) = 0 ∂λ where λ1 (c), λ2 (c) are the two positive real roots such that ∆(λ, c) = 0, and ∆(λ∗ , c∗ ) = 0,

2

∆(λ, c) = peλ

ρ−λcτ

− [cλ + δ − λ2 ]

(2.4)

(2.5)

such that for all c > c∗ , the traveling wavefront φ(x + ct) of (1.1) with the kernel function (1.2) connecting with u− and u+ exists. Furthermore, for c = c∗ , it holds that ∆(λ∗ , c) = 0, i.e., 2

peλ∗ ρ−λ∗ c∗ τ = c∗ λ∗ + δ − λ2∗

(2.6)

and for c > c∗ , it holds that ∆(λ∗ , c) < 0, i.e., 2

peλ∗ ρ−λ∗ cτ < cλ∗ + δ − λ2∗

(2.7)

for λ1 (c) < λ∗ < λ2 (c). We now define a weight function as w(x) = e−2λ∗ x ,

x∈R

where λ∗ = λ∗ (c∗ ) is the positive constant determined in Proposition 2.1. Obviously, w(x) → +∞ as x → −∞ and w(x) → 0 as x → +∞. Next, we are going to state our main result about the globally exponential stability of the traveling wavefronts of Eq. (1.1). Theorem 2.2 (Stability). Let satisfies

p δ

satisfy (2.3). For any given wavefronts φ(x + ct) with a speed c > c∗ , if c 2

eλ∗ ρ <

cλ∗ + δ − λ2∗ c∗ λ∗ + δ − λ2∗

(2.8)

the initial data holds u− ≤ u0 (s, x) ≤ u+ and the initial perturbation is u0 (s, x) − φ(x + cs) ∈ C([−τ, 0]; Hw1 (R)), then the solution of (1.1) and (1.3) satisfies u(t, x) − φ(x + ct) ∈ C([0, ∞); Hw1 (R)), u− ≤ u(t, x) ≤ u+

for (t, x) ∈ R+ × R

and ∥(u − φ)(t)∥Hw1 (R) ≤ Ce−µt ,

t≥0

for some positive constant µ. In particular, u(t, x) also converges exponential asymptotically to the wavefront φ(x+ct) in the L∞ -norm, i.e., sup |u(t, x) − φ(x + ct)| ≤ Ce−µt ,

t ≥ 0.

x∈R

Remark 2.3. (i) As is well known, in order to prove wave stability, it is often necessary to restrict the wavelength to be sufficiently small (that is, |u+ − u− | ≪ 1). Such a wave is called a weak wave; otherwise, the wave is said to be strong. If we take pδ in Proposition 2.1 to be e, that is pδ = e, so that u+ = a1 ln pδ = a1 is maximum, namely, the wavefront strength is the largest. So we prove the stability for the strong wavefronts;

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(ii) Compared with the results of Wu and Li [25], not very strong restriction to the wave speed c (i.e. it is not only for large wave speed, the result about the slower waves whose wave speed is close to the critical speed is obtained), though the kernel function (1.2) in this paper is the same as in Wu [25]; 2 cλ +δ−λ2 (iii) The condition (2.8): eλ∗ ρ < c∗ λ∗∗ +δ−λ∗2 , which is equivalent to ∗

ρ<

1 cλ∗ + δ − λ2∗ ln . 2 λ∗ c∗ λ∗ + δ − λ2∗

Here ρ is independent of c, δ, but λ∗ and c∗ are related to ρ. When c is sufficiently close to the critical cλ +δ−λ2 wave speed c∗ , then we can find that ln c∗ λ∗∗ +δ−λ∗2 ≪ 1, which means ρ need to be sufficiently small. ∗ Therefore, when ρ is small enough, we may obtain the stability for those slower waves. 3. Proof of asymptotic stability As shown in Wu and Li [25], we can similarly prove the global existence and uniqueness of the solution for the initial value problem (1.1) and (1.3). In order to prove our stability result, first of all, we need to prove the following boundedness and establish the comparison principle for Eq. (1.1), which can be obtained similarly as shown in [13,16,17], here we omit them. Lemma 3.1 (Boundedness). Let the initial data satisfy u− = 0 ≤ u0 (s, x) ≤ u+ ,

for (s, x) ∈ [−τ, 0] × R.

(3.1)

Then the solution u(t, x) of the Cauchy problem (1.1) and (1.3) satisfies u− ≤ u(t, x) ≤ u+ ,

for (t, x) ∈ [0, ∞) × R.

(3.2)

Lemma 3.2 (Comparison Principle). Let u ¯(t, x) and u(t, x) be the solutions of (1.1) and (1.2) with the initial data u ¯0 (s, x) and u0 (s, x), respectively. If u− ≤ u0 (s, x) ≤ u ¯0 (s, x) ≤ u+ ,

for (s, x) ∈ [−τ, 0] × R,

(3.3)

then u− ≤ u(t, x) ≤ u ¯(t, x) ≤ u+ ,

for (t, x) ∈ [0, ∞) × R.

(3.4)

In what follows, we are going to prove the main result, Theorem 2.2, by means of the weighted-energy method combining with comparison principle. In the proof, we will also show how to select a suitable weight function and how to get the stability of traveling wavefronts with a large initial perturbation by using the comparison principle. For given initial data u0 (s, x) satisfying u− = 0 ≤ u0 (s, x) ≤ u+ ,

for (s, x) ∈ [−τ, 0] × R,

let 

U0+ (s, x) = max{u0 (s, x), φ(x + cs)}, for (s, x) ∈ [−τ, 0] × R, U0− (s, x) = min{u0 (s, x), φ(x + cs)}, for (s, x) ∈ [−τ, 0] × R,

(3.5)

so u− ≤ U0− (s, x) ≤ u0 (s, x) ≤ U0+ (s, x) ≤ u+ , u− ≤

U0− (s, x)

≤ φ(x + cs) ≤

U0+ (s, x)

≤ u+ ,

for (s, x) ∈ [−τ, 0] × R, for (s, x) ∈ [−τ, 0] × R.

(3.6) (3.7)

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Denote U + (t, x) and U − (t, x) as the corresponding solutions of Eqs. (1.1) and (1.3) with respect to the above mentioned initial data U0+ (s, x) and U0− (s, x), i.e.,  ± 2 ±  ∂U − ∂ U + δU ± = p(g ∗ U ± )(t, x)e−a(g∗U ± )(t,x) , ∂t ∂x2  ± U (s, x) = U0± (s, x), (s, x) ∈ [−τ, 0] × R.

(t, x) ∈ R+ × R,

(3.8)

By the comparison principle, we have u− ≤ U − (t, x) ≤ u(t, x) ≤ U + (t, x) ≤ u+ , +

−

for (t, x) ∈ R+ × R,

u− ≤ U (t, x) ≤ φ(x + ct) ≤ U (t, x) ≤ u+ ,

for (t, x) ∈ R+ × R.

(3.9) (3.10)

In order to prove the stability of the traveling wavefronts presented in Theorem 2.2, we also need the following three steps as shown in [12–15]. Step 1. The convergence of U + (t, x) to φ(x + ct). Let ξ := x + ct and v0 (s, ξ) = U0+ (s, x) − φ(x + cs)

v(t, ξ) = U + (t, x) − φ(x + ct),

(3.11)

then by (3.7) and (3.10), we have v(t, ξ) ≥ 0 Let f (ω) = ωe−aω , (g ∗ φ)(ξ) =

 +∞ −∞

 

+∞

(g ∗ v)(t, ξ) =

√

−∞

−∞

√

y2

√ 1 e− 4ρ φ(ξ 4πρ

+∞

(g ∗ (φ + v))(t, ξ) =

and v0 (s, ξ) ≥ 0.

(3.12)

− cτ − y)dy, and

y2 1 e− 4ρ [φ(ξ − cτ − y) + v(t − τ, ξ − cτ − y)]dy, 4πρ

y2 1 e− 4ρ v(t − τ, ξ − cτ − y)dy. 4πρ

From Eq. (1.1), it can be verified that v(t, ξ) defined in (3.11) satisfies  vt + cvξ − vξξ + δv − pf ′ (g ∗ φ) · (g ∗ v) = pQ(t − τ, ·), v(s, ξ) = v0 (s, ξ), (s, ξ) ∈ [−τ, 0] × R,

(t, ξ) ∈ R+ × R,

(3.13)

where Q(t − τ, ·) = f (g ∗ (φ + v)) − f (g ∗ φ) − f ′ (g ∗ φ) · (g ∗ v).

(3.14)

Multiplying (3.13) by e2µt w(ξ)v(t, ξ), (µ > 0 is a positive constant to be specified later), we obtain     1 2µt 2 1 2 e wv + cwv − wvvξ e2µt + e2µt wvξ2 + e2µt w′ vvξ 2 2 t ξ   c w′ + − · + δ − µ e2µt wv 2 − pe2µt wvf ′ (g ∗ φ) · (g ∗ v) 2 w = pe2µt wv · Q(t − τ, ·). By the Cauchy–Schwarz inequality xy ≤ x2 + 14 y 2 , we have    ′ 2  w′  1 w |e2µt w′ vξ v| = e2µt w vξ · v  ≤ e2µt wvξ2 + e2µt wv 2 , w 4 w

(3.15)

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189

then (3.15) is reduced to     1 1 2µt 2 e wv + cwv 2 − wvvξ e2µt 2 2 t ξ   ′ 2  ′ c w 1 w + − · +δ−µ− e2µt wv 2 − pe2µt wvf ′ (g ∗ φ) · (g ∗ v) 2 w 4 w ≤ pe2µt wv · Q(t − τ, ·).

(3.16)

Integrating (3.16) over R × [0, t] with respect to ξ and t, we further have  ′ 2   t  ′ w 1 w −c · e2µt ∥v(t)∥2L2w + e2µs w(ξ)v 2 (s, ξ)dξds + 2δ − 2µ − w 2 w 0 R  t − 2p e2µs w(ξ)v(s, ξ)f ′ (g ∗ φ) · (g ∗ v)dξds 0

R

≤ ∥v0 (0)∥2L2w + 2p

 t 0

e2µs w(ξ)v(s, ξ)Q(s − τ, ·)dξds.

(3.17)

R

Again, using the Cauchy–Schwarz inequality we obtain |2pe2µs w(ξ)v(s, ξ)f ′ (g ∗ φ) · (g ∗ v)|  +∞ y2 1 √ e− 4ρ |v(s − τ, ξ − cτ − y)v(s, ξ)|dy ≤ 2pe2µs w(ξ)f ′ (g ∗ φ) 4πρ −∞    +∞ y2 1 1 √ ≤ pe2µs w(ξ)f ′ (g ∗ φ) e− 4ρ ηv 2 (s, ξ) + v 2 (s − τ, ξ − cτ − y) dy η 4πρ −∞   1 = pe2µs w(ξ)f ′ (g ∗ φ) ηv 2 (s, ξ) + (g ∗ v 2 ) , η for any positive constant η, which will be specified later. Thus, the third term on the left-hand-side of (3.17) is reduced to   t    e2µs w(ξ)v(s, ξ)f ′ (g ∗ φ) · (g ∗ v)dξds 2p 0 R    t 1 ≤p e2µs w(ξ)f ′ (g ∗ φ) ηv 2 (s, ξ) + (g ∗ v 2 ) dξds η 0 R  t   p t = pη e2µs w(ξ)f ′ (g ∗ φ)v 2 (s, ξ)dξds + e2µs w(ξ)f ′ (g ∗ φ) η 0 R 0 R  +∞ 2 1 − y4ρ 2 √ · e v (s − τ, ξ − cτ − y)dydξds 4πρ −∞ change of variables for the second term : ξ − cτ − y → ξ, s − τ → s, y → y  t−τ   t p e2µs = pη e2µs w(ξ)f ′ (g ∗ φ)v 2 (s, ξ)dξds + e2µτ η −τ R 0 R  +∞ y2 1 √ · e− 4ρ f ′ (g ∗ φ(ξ + cτ + y))w(ξ + cτ + y)v 2 (s, ξ)dydξds 4πρ −∞  t e2µs w(ξ)f ′ (g ∗ φ)v 2 (s, ξ)dξds ≤ pη 0

R

0

p + e2µτ η





p + e2µτ η

 t

−τ

0

e2µs ·



R

R

e2µs ·



+∞

√

−∞ +∞

−∞

√

y2 1 e− 4ρ f ′ (g ∗ φ(ξ + cτ + y))w(ξ + cτ + y)dy · v02 (s, ξ)dξds 4πρ

y2 1 e− 4ρ f ′ (g ∗ φ(ξ + cτ + y))w(ξ + cτ + y)dy · v 2 (s, ξ)dξds 4πρ

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Y.-R. Yang, L. Liu / Nonlinear Analysis 132 (2016) 183–195

 t ≤ pη

0

e2µs w(ξ)f ′ (g ∗ φ)v 2 (s, ξ)dξds

R

0



p + e2µτ η



p + e2µτ η

 t

−τ

0

e2µs ·



+∞

√

−∞

R



e2µs ·

+∞

√

−∞

R

y2 1 e− 4ρ w(ξ + cτ + y)dy · v02 (s, ξ)dξds 4πρ

y2 1 e− 4ρ w(ξ + cτ + y)dy · v 2 (s, ξ)dξds. 4πρ

(3.18)

Substituting (3.18) into (3.17) leads to  t e2µt ∥v(t)∥2L2w + e2µs Bη,µ,w (ξ)w(ξ)v 2 (s, ξ)dξds 0 R  t ≤ ∥v0 (0)∥2L2w + 2p e2µs w(ξ)v(s, ξ)Q(s − τ, ·)dξds p + e2µτ η



0

0



−τ

e2µs ·

R +∞



√

−∞

R

y 2 w(ξ + cτ + y) 1 e− 4ρ dy · w(ξ)v02 (s, ξ)dξds w(ξ) 4πρ

(by the fact that 0 ≤ f ′ (g ∗ φ(ξ + cτ + y)) ≤ 1),

(3.19)

where  2 w′ (ξ) 1 w′ (ξ) + 2δ − 2µ − − pηf ′ (g ∗ φ) w(ξ) 2 w(ξ)  +∞ y 2 w(ξ + cτ + y) p 2µτ 1 √ · e− 4ρ dy. − e η w(ξ) 4πρ −∞

Bη,µ,w (ξ) = −c ·

(3.20)

For the nonlinearity Q(t − τ, ·), using Taylor’s formula, we have Q(t − τ, ·) = f (g ∗ (φ + v)) − f (g ∗ φ) − f ′ (g ∗ φ)(g ∗ v) =

˜ f ′′ (g ∗ φ) · (g ∗ v 2 ), 2!

(3.21)

where φ˜ is some function between φ and φ + v = U + (t, x), i.e., φ ≤ φ˜ ≤ φ + v = U + (t, x). Since both φ and U + (t, x) satisfy 0 ≤ φ ≤ u+ = a1 ln pδ and 0 ≤ U + (t, x) ≤ u+ = a1 ln pδ , we then have 0 ≤ φ˜ ≤ u+ , which ˜ ˜ = −a[2 − a(g ∗ φ)]e ˜ −a(g∗φ) ensures that f ′′ (g ∗ φ) ≤ 0, (∵ 0 ≤ g ∗ φ˜ ≤ a1 ln pδ ≤ a1 ), i.e., Q(t − τ, ·) ≤ 0, which leads, with the fact v(t, ξ) ≥ 0 (see (3.12)), that  t 2p On the other hand,

w(ξ+cτ +y) w(ξ)

0

0

w(ξ)v(s, ξ)Q(s − τ, ·)dξds ≤ 0.

= e−2λ∗ (cτ +y) , it follows that +∞

 2 1 − y4ρ w(ξ + cτ + y) √ e e · dy · w(ξ)v02 (s, ξ)dξds w(ξ) 4πρ −τ R −∞   2   0   +∞ √ y √ − + 4ρλ ∗ 2 p 1 4ρ √ = e2µτ e2µs e4ρλ∗ −2λ∗ cτ  e dy  w(ξ)v02 (s, ξ)dξds η 4πρ −τ R −∞  0  2 p = e2µτ · e4ρλ∗ −2λ∗ cτ e2µs w(ξ)v02 (s, ξ)dξds η −τ R  0  2 p = e2µτ +4ρλ∗ −2λ∗ cτ e2µs w(ξ)v02 (s, ξ)dξds η −τ R  0 ≤C ∥v0 (s)∥2L2w ds.

p 2µτ e η



−τ



2µs

(3.22)

R



(3.23)

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Y.-R. Yang, L. Liu / Nonlinear Analysis 132 (2016) 183–195

Applying (3.22) and (3.23) in (3.19), we then obtain e2µt ∥v(t)∥2L2w +

 t 0

R

e2µs Bη,µ,w (ξ)w(ξ)v 2 (s, ξ)dξds ≤ ∥v0 (0)∥2L2w + C



0

−τ

∥v0 (s)∥2L2w ds.

(3.24)

Let  2 1 w′ (ξ) w′ (ξ) − pηf ′ (g ∗ φ) + 2δ − w(ξ) 2 w(ξ)  +∞ y2 w(ξ + cτ + y) p 1 √ e− 4ρ · − · dy, η −∞ w(ξ) 4πρ

Aη,w (ξ) = −c ·

(3.25)

then p Bη,µ,w (ξ) = Aη,w (ξ) − 2µ − (e2µτ − 1) · η



+∞

√

−∞

y2 w(ξ + cτ + y) 1 dy. e− 4ρ · w(ξ) 4πρ

(3.26)

Next, we will prove Bη,µ,w (ξ) > 0 by selecting the numbers η, µ and weight function w(ξ). For that purpose, we need the following lemma. 2

Lemma 3.3. Let η = e2λ∗ ρ−λ∗ cτ . Then Aη,w (ξ) ≥ C1 > 0,

ξ ∈ R.

(3.27)

+y) (ξ) = −2λ∗ and w(ξ+cτ = e−2λ∗ (y+cτ ) , we may Proof. Notice that η = e2λ∗ ρ−λ∗ cτ , w(ξ) = e−2λ∗ ξ , ww(ξ) w(ξ) obtain  +∞ y2 1 p 1 √ Aη,w (ξ) = −c · (−2λ∗ ) + 2δ − (−2λ∗ )2 − pηf ′ (g ∗ φ) − · e− 4ρ · e−2λ∗ (y+cτ ) dy 2 η −∞ 4πρ 2   +∞ √ − √y + 4ρλ∗ 1 p 4λ2∗ ρ−2λ∗ cτ 4ρ 2 √ dy e ≥ 2cλ∗ + 2δ − 2λ∗ − pη − e η 4πρ −∞ p 2 = 2cλ∗ + 2δ − 2λ2∗ − pη − e4λ∗ ρ−2λ∗ cτ η = 2cλ∗ + 2δ − 2λ2∗ − 2pη 2

′

= 2(cλ∗ + δ − λ2∗ − pη)

2

2

= 2(cλ∗ + δ − λ2∗ − peλ∗ ρ−λ∗ cτ · eλ∗ ρ )

2

= 2[cλ∗ + δ − λ2∗ − (c∗ λ∗ + δ − λ2∗ )eλ∗ ρ ] (by (2.6))   cλ∗ + δ − λ2∗ λ2∗ ρ 2 = 2(c∗ λ∗ + δ − λ∗ ) −e c∗ λ∗ + δ − λ2∗ = C1 > 0 (by (2.8)). This lemma is proved.



Lemma 3.4. Let µ1 > 0 be the unique solution of the equation C1 = 2µ + pη(e2µτ − 1). If 0 < µ < µ1 , then Bη,µ,w (ξ) ≥ C2 > 0,

ξ ∈ R.

(3.28)

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Y.-R. Yang, L. Liu / Nonlinear Analysis 132 (2016) 183–195

Proof. Applying (3.27) to (3.26), it can be examined that  +∞ y2 p 2µτ 1 √ Bη,µ,w (ξ) ≥ C1 − 2µ − (e e− 4ρ · e−2λ∗ (y+cτ ) dy − 1) · η 4πρ −∞  2  +∞ √ − √y + 4ρλ∗ 2 1 p 2µτ 4ρ √ − 1) · e4λ∗ ρ−2λ∗ cτ e = C1 − 2µ − (e dy η 4πρ −∞ 2 p = C1 − 2µ − (e2µτ − 1) · e4λ∗ ρ−2λ∗ cτ η = C1 − 2µ − pη(e2µτ − 1) = C2 > 0. This completes the proof.



Applying (3.28) to (3.24), and dropping the positive term  t e2µs Bη,µ,w (ξ)w(ξ)v 2 (s, ξ)dξds, 0

R

we then immediately establishes the first basic energy estimate as follows which is crucial for our main stability result. Lemma 3.5. It holds that e

2µt

∥v(t)∥2L2w

0

  2 ≤ C ∥v0 (0)∥L2w +

−τ



∥v0 (s)∥2L2w ds

t ≥ 0.

,

(3.29)

Similarly, differentiating Eq. (3.13) with respect to ξ, and multiplying it by e2µt w(ξ)vξ (t, ξ), and integrating the resultant equation over R × [0, t] with respect to ξ and t, then by using (3.29) in Lemma 3.5, we can obtain the second energy estimate as follows. Lemma 3.6. It holds that 0

  e2µt ∥vξ (t)∥2L2w ≤ C ∥v0 (0)∥2Hw1 +

−τ

 ∥v0 (s)∥2Hw1 ds ,

t ≥ 0.

(3.30)

t ≥ 0.

(3.31)

Therefore, (3.29) and (3.30) imply the following fact. Lemma 3.7. It holds that ∥v(t)∥2Hw1

−2µt

≤ Ce



∥v0 (0)∥2Hw1

0

 +

−τ



∥v0 (s)∥2Hw1 ds

,

Notice that w(ξ) → 0 as ξ → ∞, we cannot conclude Hw1 (R) ↩→ C(R). However, for any interval I = (−∞, ξ∗ ] for some large ξ∗ ≫ 1, there is the Sobolev’s embedding result Hw1 (I) ↩→ C(I), which can be combined with (3.31) and be given the following L∞ -estimate. Lemma 3.8. It holds that −µt

sup |v(t, ξ)| ≤ Ce ξ∈I



∥v0 (0)∥2Hw1



0

+ −τ

 21

∥v0 (s)∥2Hw1 ds

,

t ≥ 0,

(3.32)

for any interval I = (−∞, ξ∗ ] with some large ξ∗ ≫ 1. However, we need the L∞ -convergence in (3.32) in the whole space (−∞, +∞). Thus, we are going to prove the convergence at ξ = +∞.

Y.-R. Yang, L. Liu / Nonlinear Analysis 132 (2016) 183–195

193

Lemma 3.9. It holds that lim |v(t, ξ)| ≤ Ce−µ2 t ,

t ≥ 0,

ξ→+∞

where µ2 > 0 and µ2 satisfies δ − µ2 − peµ2 τ f ′ (u+ ) > 0. Proof. As shown in (3.22), for the sake of Q(t − τ, ·) ≤ 0 and v(t, ξ) ≥ 0, (3.13) is reduced to  +∞ y2 ∂v ∂v ∂ 2 v 1 √ +c − 2 + δv − pf ′ (g ∗ φ) · e− 4ρ v(t − τ, ξ − cτ − y)dy ≤ 0. ∂t ∂ξ ∂ξ 4πρ −∞

(3.33)

Taking limits as ξ → +∞, and noting that vξ (t, +∞) = 0, vξξ (t, +∞) = 0 due to the boundedness of v(t, ξ) for all ξ ∈ R, we obtain  +∞ y2 d 1 √ e− 4ρ dy ≤ 0. v(t, ∞) + δv(t, ∞) − pf ′ (u+ )v(t − τ, ∞) dt 4πρ −∞ Since

 +∞ −∞

y2

√ 1 e− 4ρ dy 4πρ

= 1, therefore d v(t, ∞) + δv(t, ∞) − pf ′ (u+ )v(t − τ, ∞) ≤ 0. dt

(3.34)

Multiplying (3.34) by eµ2 t (µ2 is a positive constant to be specified later) and integrating it over [0, t], we have  t  t  t d eµ2 s v(s, ∞)ds + δ eµ2 s v(s, ∞)ds − pf ′ (u+ ) eµ2 s v(s − τ, ∞)ds ≤ 0. (3.35) ds 0 0 0 For the first term in (3.35), we get  t  t d eµ2 s v(s, ∞)ds. eµ2 s v(s, ∞)ds = eµ2 t v(t, ∞) − v0 (0, ∞) − µ2 ds 0 0

(3.36)

By the change of variable s − τ → s for the third term in (3.35), we obtain  t  t−τ pf ′ (u+ ) eµ2 s v(s − τ, ∞)ds = pf ′ (u+ ) eµ2 (s+τ ) v(s, ∞)ds 0

−τ

≤ pf ′ (u+ )eµ2 τ



= pf ′ (u+ )eµ2 τ



t

eµ2 s v(s, ∞)ds

−τ 0

eµ2 s v(s, ∞)ds

−τ ′

+ pf (u+ )e

t



µ2 τ

0

eµ2 s v(s, ∞)ds.

(3.37)

Substituting (3.37) and (3.36) into (3.35), we have  t  t eµ2 t v(t, ∞) − v0 (0, ∞) − µ2 eµ2 s v(s, ∞)ds + δ eµ2 s v(s, ∞)ds 0

− pf ′ (u+ )eµ2 τ



0

0

eµ2 s v(s, ∞)ds − pf ′ (u+ )eµ2 τ

−τ

 0

t

eµ2 s v(s, ∞)ds ≤ 0,

(3.38)

i.e. eµ2 t v(t, ∞) + [δ − µ2 − pf ′ (u+ )eµ2 τ ]

 0

t

eµ2 s v(s, ∞)ds ≤ C3 ,

(3.39)

194

Y.-R. Yang, L. Liu / Nonlinear Analysis 132 (2016) 183–195

0 where C3 = v0 (0, ∞) + pf ′ (u+ )eµ2 τ −τ eµ2 s v(s, ∞)ds. Moreover, we can easily check 0 ≤ f ′ (u+ ) < pδ , therefore δ − pf ′ (u+ ) > 0. Thus, there exists a small µ2 > 0 such that δ − µ2 − pf ′ (u+ )eµ2 τ > 0, and (3.39) yields v(t, ∞) ≤ Ce−µ2 t . This completes the proof.



Remark 3.10. There is another method for the proof of Lemma 3.9, we can use the result of Halanay’s inequality (or called Hanal’s inequality or Halanay lemma) for (3.34), which immediately gives the desired exponential decay estimate |v(t, ∞)| ≤ Ce−µ2 t , where µ2 = ε(δ − pf ′ (u+ )) and 0 < ε < 1, and the proof of (3.34) below is not necessary in the first method, please see Lemma 3.8–3.10 in Mei’s [11]. For more papers about Halanay’s inequality, you can refer to [2,5–7]. Combining Lemmas 3.8 and 3.9, and taking 0 < µ < min{µ1 , µ2 }, we prove the L∞ - convergence in Theorem 2.2 for all ξ ∈ R, i.e., Lemma 3.11. It holds that sup |U + (t, x) − φ(x + ct)| ≤ Ce−µt ,

t ≥ 0,

ξ∈R

where 0 < µ < min{µ1 , µ2 }. Step 2. The convergence of U − (t, x) to φ(x + ct). Let ξ := x + ct and v(t, ξ) = φ(x + ct) − U − (t, x),

v0 (s, ξ) = φ(x + cs) − U0− (s, x).

As shown in the process of Step 1, we can similarly prove the convergence of U − (t, x) to φ(x + ct), i.e., Lemma 3.12. It holds that sup |U − (t, x) − φ(x + ct)| ≤ Ce−µt ,

t ≥ 0,

ξ∈R

where 0 < µ < min{µ1 , µ2 }. Step 3. The convergence of u(t, x) to φ(x + ct). In this step, we are going to prove Theorem 2.2, namely, Lemma 3.13. It holds that sup |u(t, x) − φ(x + ct)| ≤ Ce−µt ,

t≥0

ξ∈R

for 0 < µ < min{µ1 , µ2 }. + Proof. Since the initial data satisfy U0− (s, x) ≤ u− 0 (s, x) ≤ U0 (s, x), by Lemma 3.2, it can be proved that − the corresponding solutions of (1.1) and (1.2) satisfy U (t, x) ≤ u(t, x) ≤ U + (t, x), (t, x) ∈ R+ × R. Thanks to Lemmas 3.11, 3.12, and using the squeeze theorem, we finally prove

sup |u(t, x) − φ(x + ct)| ≤ Ce−µt , ξ∈R

where 0 < µ < min{µ1 , µ2 }. This completes the proof.



t≥0

Y.-R. Yang, L. Liu / Nonlinear Analysis 132 (2016) 183–195

195

Acknowledgments The first author was supported by the NSF of China (11301241), Science and Technology Plan Foundation of Gansu Province of China (145RJYA250), Institutions of higher learning scientific research project of Gansu Province of China (2013A-044) and Young Scientist Foundation of Lanzhou Jiaotong University of China (2011029). References [1] X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations 2 (1997) 125–160. [2] K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. [3] S.A. Gourley, Traveling fronts in the diffusive Nicholson’s blowflies equation with distributed delays, Math. Comput. Modelling 32 (2000) 843–853. [4] W.S.C. Gurney, S.P. Blythe, R.M. Nisbet, Nicholson’s blowflies revisited, Nature 287 (1980) 17–21. [5] A. Halanay, Differential Equation: Stability, Oscillations, Time Lags, Academic Press, New York, London, 1966. [6] A. Halanay, J.A. Yorke, Some new results and problems in the theory of differential-delay equations, SIAM Rev. 13 (1971) 55–80. [7] A. Ivanov, E. Liz, S. Trofimchuk, Halanay inequality, York 3/2 stability criterion, and differential equations with maxima, Tohoku Math. J. 54 (2002) 277–295. [8] G.R. Li, M. Mei, Y.S. Wong, Nonlinear stability of traveling wavefronts in an age-structured reaction–diffusion population model, Math. Biosci. Eng. 5 (2008) 85–100. [9] W.T. Li, S. Ruan, Z.C. Wang, On the diffusive Nicholson’s blowflies equation with nonlocal delays, J. Nonlinear Sci. 17 (2007) 505–525. [10] G.J. Lin, Traveling waves in the Nicholson’s blowflies equation with spatio-temporal delay, Appl. Math. Comput. 209 (2009) 314–326. [11] C.K. Lin, C.T. Lin, Y. Lin, M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson’s blowflies equation, SIAM J. Math. Anal. 46 (2014) 1053–1084. [12] C.-K. Lin, M. Mei, On traveling wavefronts of Nicholson’s blowflies equations with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010) 135–152. [13] M. Mei, Stability of traveling wavefronts for time-delay reaction–diffusion equations, in: Proceedings of the 7th AIMS International Conference, Texas, USA, Discrete Cont. Dyn. Syst., Supplement, 2009, pp. 526–535. [14] M. Mei, C.-K. Lin, C.-T. Lin, J.W.-H. So, Traveling wavefronts for time-delayed reaction–diffusion equation: (I) Local nonlinearity, J. Differential Equations 247 (2009) 495–510. [15] M. Mei, C.-K. Lin, C.-T. Lin, J.W.-H. So, Traveling wavefronts for time-delayed reaction–diffusion equation: (II) Nonlocal nonlinearity, J. Differential Equations 247 (2009) 511–529. [16] M. Mei, J.W.-H. So, Stability of strong traveling waves for a nonlocal time-delayed reaction–diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 551–568. [17] M. Mei, J.W.-H. So, M.Y. Li, S.S. Shen, Asymptotic stability of traveling waves for the Nicholson’s blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579–594. [18] D.H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math. 22 (1976) 312–355. [19] K.W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc. 302 (1987) 587–615. [20] H. Smith, X.Q. Zhao, Global asymptotic stability of the traveling waves in delayed reaction–diffusion equations, SIAM J. Math. Anal. 31 (2000) 514–534. [21] J.W.-H. So, J. Wu, Y. Yang, Numerical Hopf bifurcation analysis on the diffusive Nicholson’s blowflies equation, Appl. Math. Comput. 111 (2000) 53–69. [22] J.W.-H. So, Y. Yang, Dirichlet problem for Nicholson’s blowflies equation, J. Differential Equations 150 (1998) 317–348. [23] J.W.-H. So, X. Zou, Traveling waves for the diffusive Nicholson’s blowflies equation, Appl. Math. Comput. 122 (2001) 385–392. [24] Z.C. Wang, W.T. Li, S. Ruan, Traveling wave fronts in reaction–diffusion systems with spatio-temporal delays, J. Differential Equations 222 (2006) 185–232. [25] S.L. Wu, W.T. Li, Exponential stability of traveling fronts in monostable reaction–advection–diffusion equations with nonlocal delay, Discrete Contin. Dyn. Syst. Ser. B 17 (2012) 347–366.