# Global Asymptotic Stability in a Class of Reaction-Diffusion Equations with Time Delay

Authors: Yueding Yuan; Zhiming Guo
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We study a very general class of delayed reaction-diffusion equations in which the reaction term can be nonmonotone and spatially nonlocal. By using a fluctuation method, combined with the careful analysis of the corresponding characteristic equations, we obtain some sufficient conditions for the global asymptotic stability of the trivial solution and the positive steady state to the equations subject to the Neumann boundary condition.


---

## Body

## 1. Introduction

There has been a growing interest in the dynamic behavior of spatial nonlocal and time-delayed population systems since the 1970s [1]. When the death function of such a system is linear many researchers used the theory of monotone semiflows, the comparison arguments, and the fluctuation method to study spreading speeds, traveling waves, and the global stability (see, e.g., [2–10]). However, researches on these problems become relatively rare for nonmonotone delayed reaction-diffusion systems in which the death function is nonlinear (see [11, 12]). The reason lies in the fact that it is difficult to establish an appropriate expression for solutions to study the solution semiflow under this case.In this paper, we will investigate the global asymptotic stability of the positive steady state for the following time-delayed reaction-diffusion equation:(1)∂w(t,x)∂t=dΔw(t,x)-f(w(t,x))+∫Ω‍k(α,x,y)b(w(t-τ,y))dy,∂w(t,x)∂n=0,t>0,x∈∂Ω,w(t,x)=ϕ(t,x)≥0,t∈[-τ,0],x∈Ω,
where d>0, α≥0, τ≥0, Δ denotes the Laplacian operator on ℝm, Ω is a bounded and open domain of ℝm with a smooth boundary ∂Ω, ∂/∂n is the differentiation in the direction of the outward normal n to ∂Ω, and the kernel function k(α,x,y) is given by
(2)k(α,x,y)={∑n=1+∞‍e-λnαφn(x)φn(y),ifα>0,δ(x-y),ifα=0.
Here, 0=λ1<λ2≤⋯≤λn≤⋯ with lim⁡n→∞λn=+∞ is the eigenvalue of the linear operator -Δ subject to the homogeneous Neumann boundary condition on ∂Ω, φn is the eigenvector corresponding to λn, {φn}n=1+∞ is a complete orthonormal system in the space L2(Ω¯), φ1(x)>0 for all x∈Ω, and δ(x) is the Dirac function on ℝm [10, 13]. Throughout this paper, we assume that the functions f and b satisfy the following.(A1)
b:ℝ+→ℝ+ is Lipschitz continuous with b(0)=0 and b′(0)>0, and b(w)≤b′(0)w for all w≥0.(A2)
f(w)=wg(w) for all w≥0, where g:ℝ+→ℝ+ is Lipschitz continuous with g′(0)≥0, g(w)>0, and g′(w)≥0 for all w>0.(A3)
There exists a positive numberM such that, for all w>M, b¯(w)<f(w), where b¯(w)=max⁡u∈[0,w]b(u).In the monotone case, where the functionb(w) increases with w>0, Xu and Zhao [12] studied the global dynamics of (1) and obtained some results on the uniqueness and global attractivity of a positive steady state by using the theory of monotone dynamical systems. In the case of f(w)=μw, Zhao [10] proved the global attractivity of the positive constant equilibrium for (1) by using a fluctuation method of Thieme and Zhao [14], where μ is a positive constant. In the case where f(w)=μw and α=0, Yi and Zou [7] proved the global attractivity of the unique positive constant equilibrium for (1) by combining a dynamical systems argument and some subtle inequalities. In the case where Ω=[0,L] and f(w)=μw, (1) reduces to the equation derived in [3], where the numerical solutions are considered. A global convergence theorem was obtained in [11] for a special case of (1).The aim of this paper is to establish some criteria to guarantee the global asymptotic stability of the trivial solution and the positive steady state for (1) by using a fluctuation method, combined with the careful analysis of the corresponding characteristic equations. The interesting thing is that main results obtained in this paper extend the related existing results.The rest of this paper is organized as follows. We will present some preliminary results in Section2. Our main results are presented and proved in Sections 3 and 4, where we obtain sufficient conditions to ensure the global asymptotic stability of the trivial solution and the positive steady state for (1) in a nonmonotone case. In Section 5, we provide four examples to illustrate the applicability of the main results.


## 2. Preliminaries

Firstly, we show that the kernelk(α,x,y) in (2) enjoys the following properties.Lemma 1.
Forα>0, one has (i)
(∂/∂n)k(α,x,y)|x∈∂Ω=(∂/∂n)k(α,x,y)|y∈∂Ω=0,(ii)
0<k(α,x,y)≤C*, for all x,y∈Ω, where C* is a positive constant depending only on m and Ω,(iii)
|φn(x)|≤C*exp⁡[(1/2)λnα], for all x∈Ω, n=1,2,…,(iv)
∫Ω¯‍k(α,x,y)dy=1, for all x∈Ω.Proof.
The verification of (i) is straightforward and is thus omitted. Part (ii) follows from [15, Lemma 3.2.1 and Theorem 4.4.6] since k(α,x,y) is a heat kernel of the heat equation (Δ-(∂/∂α))u(x,α)=0. Part (iii) follows from e-λnα(φn(x))2≤k(α,x,x)≤C*, for all x∈Ω, n=1,2,…. And part (iv) follows from λ1=0, φ1(x)≡1/mes(Ω), and ∫Ω¯‍φn(y)dy=0 for all n=2,3,…, where mes(Ω) is the measure of Ω. The proof is completed.Let𝕏=C(Ω¯,ℝ) and 𝕏+={ϕ∈𝕏∣ϕ(x)≥0,∀x∈Ω¯}. Then (𝕏,𝕏+) is a strongly ordered Banach space. It is well known that the differential operator A=dΔ generates a C0-semigroup T(t) on 𝕏. Moreover, the standard parabolic maximum principle (see, e.g., [16, Corollary 7.2.3]) implies that the semigroup T(t):𝕏→𝕏 is strongly positive in the sense that T(t)(𝕏+∖{0})⊂Int(𝕏+),∀t>0.Let𝕐=C([-τ,0],𝕏) and 𝕐+=C([-τ,0],𝕏+). For the sake of convenience, we will identify an element ϕ∈𝕐 as a function from [-τ,0]×Ω¯ to ℝ defined by ϕ(s,x)=ϕ(s)(x), and for each s∈[-τ,0], we regard f(ϕ(s)) as a function on Ω¯ defined by f(ϕ(s))=f(ϕ(s,·)). For any function w(·):[-τ,σ)→𝕏, where σ>0, we define wt∈𝕐, t∈[0,σ) by wt(s)=w(t+s),∀s∈[-τ,0]. Define F:𝕐+→𝕏 by
(3)F(ϕ)(x)=-f(ϕ(0,x))+∫Ω¯‍k(α,x,y)b(ϕ(-τ,y))dy,∀x∈Ω¯,ϕ∈𝕐+.
Then we can rewrite (1) as an abstract functional equation:
(4)dw(t)dt=Aw(t)+F(wt),t≥0,w0=ϕ∈𝕐+.
Therefore, we can write (4) as an integral equation:
(5)w(t)=T(t)ϕ(0)+∫0t‍T(t-s)F(ws)ds,t≥0,w0=ϕ∈𝕐+,
whose solutions are called mild solutions for (1).SinceT(t):𝕏→𝕏 is strongly positive, we have
(6)lim⁡h→0+dist⁡(ϕ(0)+hF(ϕ),𝕏+)=0,∀ϕ∈𝕐+.
By [17, Proposition 3 and Remark 2.4] (or [18, Corollary 8.1.3]), for each ϕ∈𝕐+, (1) has a unique noncontinuable mild solution w(t,ϕ) with w0=ϕ, and w(t,ϕ)∈𝕏+ for all t∈(0,σϕ). Moreover, w(t,ϕ) is a classical solution of (1) for t>τ (see [18, Corollary 2.2.5]).By the same arguments as in the proof of [12, Theorems 2.1 and 3.1], we have the following two lemmas.Lemma 2.
Let (A1)–(A3) hold. Then, for eachϕ∈𝕐+, a unique solution w(t,ϕ) of (1) globally exists on [-τ,∞), limsup⁡t→∞w(t,x,ϕ)≤M uniformly for x∈Ω¯, and the solution semiflow Φ(t)=wt(·):𝕐+→𝕐+, t≥0, admits a connected global attractor.Lemma 3.
Let (A1)–(A3) hold, and letw(t,x,ϕ) be the solution of (1) with ϕ∈𝕐+. Then the following two statements are valid. (i)
Ifb′(0)<g(0), then for any ϕ∈𝕐+, we have limsup⁡t→∞w(t,x,ϕ)=0 uniformly for x∈Ω¯.(ii)
Ifb′(0)>g(0), then (1) admits at least one spatially homogeneous steady state w*∈(0,M], and there exists η>0 such that for any ϕ∈𝕐+ with ϕ(0,·)≢0 we have liminf⁡t→∞w(t,x,ϕ)≥η uniformly for x∈Ω¯.Note that in case (ii) above, the functionS(w)=b(w)-f(w) satisfies S(0)=0, S′(0)>0, and S(M)≤0. Therefore, there exists at least one positive number w*∈(0,M] such that S(w*)=0, and hence, w* is a spatially homogeneous steady state of (1).


## 3. Global Attractivity

In this section, we establish the global attractivity of the positive and spatially homogeneous steady statew* for (1) by the fluctuation method used in [10, Theorem 3.1].Motivated by [10, Section 3], we assume further that the functions f(w) and b(w) satisfy the following.(A4)
b′(0)>g(0),(b(w)/f(w)) is strictly decreasing for w∈(0,M], and f(w) and b(w) have the property (P) that, for any u,v∈(0,M] satisfying u≤w*≤v, f(u)≥b(v), and f(v)≤b(u), we have u=v.Note that ifb(w) is nondecreasing for w∈[0,M], then f(w) and b(w) have the property (P). Indeed, for any 0<u≤w*≤v≤M with f(u)≥b(v) and f(v)≤b(u), we have
(7)f(w*)≤f(v)≤b(u)≤b(w*)≤b(v)≤f(u)≤f(w*),
which implies that u=v=w*. Combining this observation and [10, Lemma 3.1] with μw replaced by f(w), where μ>0, we then have the following result.Lemma 4.
Either of the following two conditions is sufficient for the property (P) in condition (A4) to hold.(P1)
b(w) is nondecreasing for w∈[0,M].(P2)
f(w)b(w) is strictly increasing for w∈(0,M].Now we are in a position to prove our main result in this section.Theorem 5.
Assume that (A1)–(A4) hold, and letw(t,x,ϕ) be the solution of (1) with ϕ∈𝕐+. Then for any ϕ∈𝕐+ with ϕ(0,·)≢0, we have lim⁡t→∞w(t,x,ϕ)=w* uniformly for x∈Ω¯.In order to prove Theorem5, we will need the following lemma.Lemma 6.
Assume that (A1)–(A3) hold, and letw(t,x)≡w(t,x,ϕ) be the solution of (1) with ϕ∈𝕐+. Then w(t,x) satisfies
(8)w(t,x)=e-γt∫Ω‍k(dt,x,y)ϕ(0,y)dy+∫0t‍e-γs∫Ω‍k(ds,x,y)×[∫Ω‍γw(t-s,y)-f(w(t-s,y))+∫Ω‍k(α,y,z)b(w(t-s-τ,z))dz]dyds,
where γ=max⁡w∈[0,M]f′(w) and the kernel function k is given in (2).Proof.
Let(9)H(t,x)≡γw(t,x)-f(w(t,x))+∫Ω‍k(α,x,y)b(w(t-τ,y))dy.
Since 𝕏⊂L2(Ω¯), for each t≥0, there exist real numbers an(t) and bn(t), n=1,2,…, such that
(10)w(t,x)=∑n=1+∞‍an(t)φn(x),(11)H(t,x)=∑n=1+∞‍bn(t)φn(x).
Therefore, by (10), (11), and (1), we have
(12)an(0)=∫Ω‍ϕ(0,y)φn(y)dy,bn(s)=∫Ω‍H(s,y)φn(y)dy,(13)dan(t)dt=-(dλn+γ)an(t)+bn(t),n=1,2,….
By using the variation of constants method, we obtain
(14)an(t)=[an(0)+∫0t‍e(dλn+γ)sbn(s)ds]e-(dλn+γ)t,n=1,2,….
Thus, by (10), (12), and (14), we further get
(15)w(t,x)=∑n=1+∞‍[an(0)+∫0t‍e(dλn+γ)sbn(s)ds]e-(dλn+γ)tφn(x)=e-γt∑n=1+∞‍∫Ω‍ϕ(0,y)e-dλntφn(x)φn(y)dy+∫0t‍e-γ(t-s)∑n=1+∞‍∫Ω‍e-dλn(t-s)H(s,y)φn(x)φn(y)dyds=e-γt∫Ω‍ϕ(0,y)k(dt,x,y)dy+∫0t‍e-γ(t-s)∫Ω‍H(s,y)k(d(t-s),x,y)dyds=e-γt∫Ω‍ϕ(0,y)k(dt,x,y)dy+∫0t‍e-γs∫Ω‍H(t-s,y)k(ds,x,y)dyds.
Therefore, (8) follows immediately from (9) and (15). The proof is completed.Proof of Theorem5.
For any givenϕ∈𝕐+ with ϕ(0,·)≢0, let ω(ϕ) be the omega limit set of the positive orbit through ϕ for the solution semiflow Φ(t). By Lemma 1, we get ω(ϕ)⊂𝔸⊆𝕐[0,M], where 𝔸 is the global attractor of the solution semiflow Φ(t) and
(16)𝕐[0,M]≡{ϕ∈𝕐∣0≤ϕ(θ,x)≤M,∀(θ,x)∈[-τ,0]×Ω¯}.
Note that 𝔸 is a maximal compact invariant set of the solution semiflow Φ(t). Thus, it is sufficient to prove the global attractivity of w* for all ϕ∈𝕐[0,M] with ϕ(0,·)≢0.
Letϕ∈𝕐[0,M] be given such that ϕ(0,·)≢0. Then it follows from Lemma 6 that
(17)w(t,x)=e-γt∫Ω‍k(dt,x,y)ϕ(0,y)dy+∫0t‍e-γs∫Ω‍k(ds,x,y)×[γw(t-s,y)-f(w(t-s,y))∫Ω‍+∫Ω‍k(α,y,z)b(w(t-s-τ,z))dz]dyds,
where w(t,x)≡w(t,x,ϕ) is the solution of (1) starting from the initial function ϕ. Following [19], we define a function h:[0,M]×[0,M]→ℝ by
(18)h(u,v)={min⁡{b(w)∣u≤w≤v},ifu≤v,max⁡{b(w)∣v≤w≤u},ifv≤u.
Then h(u,v) is nondecreasing in u∈[0,M] and nonincreasing in v∈[0,M]. Moreover, b(w)=h(w,w), ∀w∈[0,M], and h(u,v) is continuous in (u,v)∈[0,M]×[0,M] (see [20, Section 2]). Therefore, by (17), we have
(19)w(t,x)=e-γt∫Ω‍k(dt,x,y)ϕ(0,y)dy+∫0t‍e-γs∫Ω‍k(ds,x,y)×[∫Ω‍γw(t-s,y)-f(w(t-s,y))+∫Ω‍k(α,y,z)×h(w(t-s-τ,z),w(t-s-τ,z))dz∫Ω‍]dyds.
Let
(20)w∞(x)≡limsup⁡t→∞w(t,x),w∞(x)≡liminf⁡t→∞w(t,x),∀x∈Ω¯.
Then Lemmas 2 and 3 imply that
(21)M≥w∞(x)≥w∞(x)≥η>0,∀x∈Ω¯.
On the other hand, note that γ=max⁡w∈[0,M]f′(w). Therefore, the function γw-f(w) is nondecreasing in w∈[0,M]. Thus, by Fatou’s lemma and (19), we further get
(22)w∞(x)≤∫0∞‍e-γs∫Ω‍k(ds,x,y)×[∫Ω‍γw∞(y)-f(w∞(y))+∫Ω‍k(α,y,z)h(w∞(z),w∞(z))dz]dyds.
Let
(23)w∞≡sup⁡x∈Ω¯w∞(x),w∞≡inf⁡x∈Ω¯w∞(x).
Then
(24)M≥w∞≥w∞≥η>0.
Moreover, it follows from Lemma 1 that
(25)∫Ω‍k(ds,x,y)dy=1,∫Ω‍k(α,x,y)dy=1,∀s≥0,x∈Ω¯.
Therefore, by (22), we have
(26)w∞≤[γw∞-f(w∞)+h(w∞,w∞)]∫0∞‍e-γsds=1γ[γw∞-f(w∞)+h(w∞,w∞)].
Thus,
(27)f(w∞)≤h(w∞,w∞).
Similarly, we have
(28)f(w∞)≥h(w∞,w∞).
By (18), we may find u,v∈[w∞,w∞]⊂(0,M] such that
(29)h(w∞,w∞)=b(u),h(w∞,w∞)=b(v).
It then follows from (27) and (28) that
(30)b(u)≥f(w∞)≥f(u),b(v)≤f(w∞)≤f(v),
and hence,
(31)b(v)f(v)≤1=b(w*)f(w*)≤b(u)f(u).
This, together with the strict monotonicity of b(w)/f(w) for w∈(0,M], implies that u≤w*≤v. Moreover, by (27) and (28), we also have
(32)b(u)≥f(w∞)≥f(v),b(v)≤f(w∞)≤f(u).
Therefore, the property (P) implies that
(33)u=v=w*.
Thus, by (30), we obtain
(34)w∞=w∞=w*.
Since
(35)w∞≥w∞(x)≥w∞(x)≥w∞,x∈Ω¯,
we further get
(36)w∞(x)=w∞(x)=w*,x∈Ω¯.
This implies that
(37)lim⁡t→∞w(t,x)=w*,x∈Ω¯.
It remains to prove thatlim⁡t→∞w(t,x)=w* uniformly for x∈Ω¯. For any ψ∈ω(ϕ), there exists a sequence tn→∞ such that Φ(tn)ϕ→ψ in 𝕐 as n→∞. Therefore, we have
(38)lim⁡n→∞w(tn+θ,x,ϕ)=ψ(θ,x)
uniformly for (θ,x)∈[-τ,0]×Ω¯. By (37), we further get
(39)ψ(θ,x)=w*,∀(θ,x)∈[-τ,0]×Ω¯.
Thus, we obtain ω(ϕ)={w*}, which implies that w(t,·,ϕ) converges to w* in 𝕏 as t→∞. The proof is completed.


## 4. Global Asymptotic Stability

In this section, we establish the global asymptotic stability of the trivial solution and the positive and spatially homogeneous steady statew* for (1) by the careful analysis of the corresponding characteristic equations. To this end, we first give the following formal definitions of stability (see, e.g., [18, Remark 2.1.3]).Definition 7.
Letw=w^ be a steady state of the abstract equation (4). It is called stable if for any ε>0 there exists δ>0 such that the solution w(t,ϕ) of (4) with ∥ϕ-w^∥𝕐<δ satisfies ∥w(t,ϕ)-w^∥𝕏<ε, for all t≥0. It is called unstable if it is not stable. It is asymptotically stable if it is stable and there exists δ0>0 such that the solution w(t,ϕ) of (4) with ∥ϕ-w^∥𝕐<δ0 satisfies lim⁡t→+∞∥w(t,ϕ)-w^∥𝕏=0. It is globally asymptotically stable if it is stable and any solution w(t,ϕ) of (4) with arbitrary ϕ∈𝕐 satisfies lim⁡t→+∞∥w(t,ϕ)-w^∥𝕏=0.Letw^ be a spatially homogeneous steady state for (1) (e.g., the trivial solution and w*). Define G:𝕐+→𝕏 by
(40)G(ϕ)(x)=-f′(w^)ϕ(0,x)+b′(w^)∫Ω¯‍k(α,x,y)ϕ(-τ,y)dy,∀x∈Ω¯,ϕ∈𝕐+,
where f′(w^)=(df(w)/dw)|w=w^ and b′(w^)=(db(w)/dw)|w=w^. Note that k(α,x,y) is given in (2). Then we can write the linearized equation of (1) at w=w^ as the following abstract functional equation
(41)dw(t)dt=Aw(t)+G(wt),t≥0,w0=ϕ∈𝕐+,
where A can be referred to Section 2.For each complex numberλ we define the 𝕏-valued linear operator Θ(λ) by
(42)Θ(λ)u=Au-λu+G(eλ·u),u∈Dom⁡(A),
where eλ·u∈𝕐 is defined by (note that we use 𝕐 to denote its complexification here)
(43)(eλ·u)(θ)=eλθu,θ∈[-τ,0].
We will call λ a characteristic value of (41) if there exists u∈Dom⁡(A)∖{0} solving the characteristic equation Θ(λ)u=0 (see, e.g., [18]). Since Dom⁡(A)⊂𝕏⊂L2(Ω¯), for any u∈Dom⁡(A)∖{0}, there exist complex numbers an,n=1,2,…, such that
(44)u(x)=∑n=1+∞‍anφn(x).
Therefore, by (2), (42), and (44), we have
(45)Θ(λ)u(x)=dΔu(x)-λu(x)-f′(w^)u(x)+b′(w^)∫Ω¯‍e-τλu(y)k(α,x,y)dy=∑n=1+∞‍an[-dλn-λ-f′(w^)+b′(w^)e-τλe-λnα]φn(x).
Thus, the characteristic value λ of (41) satisfies at least one of the following equations:
(46)λ=-dλn-f′(w^)+b′(w^)e-λnαe-τλ,n=1,2,….Lemma 8.
Assume that (A1)–(A3) hold, and letβ be the smallest real number such that if λ is a characteristic value of (41), then Reλ≤β. One has the following: (i)
ifb′(w^)>f′(w^), then β>0,(ii)
if-f′(w^)≤b′(w^)<f′(w^), then β<0,(iii)
ifb′(w^)=f′(w^), then β=0.Proof.
(i) Ifb′(w^)>f′(w^), then, by (46) and [21, Proposition 4.6], there exists at least one characteristic value λ of (41) such that Reλ>0. Therefore, β>0.
(ii) If-f′(w^)≤b′(w^)<f′(w^), then since 0=λ1<λ2≤⋯≤λn≤⋯, we have
(47)-[dλn+f′(w^)]eλnα≤b′(w^)<[dλn+f′(w^)]eλnα,n=1,2,….
Therefore, by (46) and [21, Proposition 4.6], all the characteristic values of (41) have negative real parts. Thus, it follows from [18, Theorem 3.1.10] that β<0.
(iii) Ifb′(w^)=f′(w^), then λ=0 is a characteristic value of (41). Therefore, β≥0. If β>0, then there exists at least one characteristic value of (41) λ(0) and a positive number n such that Reλ(0)>0 and
(48)λ(0)=-dλn-f′(w^)+b′(w^)e-λnαe-τλ(0).
Let λ(0)=x(0)+iy(0), where x(0) and y(0) both are real numbers. Then x(0)>0. By (48), we have
(49)x(0)=-dλn-f′(w^)+f′(w^)e-λnαe-τx(0)cos⁡(τy(0)),
and hence, cos⁡(τy(0))>0. This implies that
(50)x(0)≤-f′(w^)+f′(w^)e-τx(0)cos⁡⁡(τy(0))=f′(w^)[e-τx(0)cos⁡(τy(0))-1].
But, since x(0)>0, we have e-τx(0)cos⁡(τy(0))<1. Therefore,
(51)x(0)≤f′(w^)[e-τx(0)cos⁡(τy(0))-1]<0,
contradicting x(0)>0. This contradiction proves β=0. The proof is completed.Now we are ready to summarize our main results on the global stability. By Definition7, Lemmas 3 and 8, Theorem 5, [18, Corollary 3.1.11], and the principle of linearized stability (see, e.g., [21]), we obtain the following.Theorem 9.
Assume that (A1)–(A3) hold. Then the following two statements are valid.(i)
Ifb′(0)<g(0), then the zero solution of (1) is globally asymptotically stable in 𝕐+.(ii)
Ifb′(0)>g(0), then the zero solution of (1) is unstable, and (1) admits at least one spatially homogeneous steady state w*∈(0,M].Theorem 10.
Assume that (A1)–(A3) hold, andb′(0)>g(0). Then the following two statements for the positive and spatially homogeneous steady state w* of (1) are valid. (i)
Ifb′(w*)>f′(w*), then w* is unstable.(ii)
If-f′(w*)≤b′(w*)<f′(w*) and (A4) hold, then w* is globally asymptotically stable in 𝕐+∖{0}.


## 5. Examples

In this section, we present four examples to illustrate the feasibility of our main results.Example 1.
Consider the equation resulting from lettingf(w)=μw in (1); that is,
(52)∂w(t,x)∂t=dΔw(t,x)-μw(t,x)+∫Ω‍k(α,x,y)b(w(t-τ,y))dy,∂w(t,x)∂n=0,t>0,x∈∂Ω,w(t,x)=ϕ(t,x)≥0,t∈[-τ,0],x∈Ω,
where μ is a positive constant.In this case, we now formulate the following assumptions to replace (A2)–(A4):(A2′)
There exists a positive numberM such that, for all w>M, b¯(w)<μw, where b¯(w)=max⁡u∈[0,w]b(u).(A3′)
b′(0)>μ, (b(w)/w) is strictly decreasing for w∈(0,M], and b(w) has the property (P′) that, for any u,v∈(0,M] satisfying u≤w*≤v, μu≥b(v), and μv≤b(u), we have u=v.By applying Theorems9 and 10, we then obtain the following results for (52).Theorem 11.
Assume that (A1) and (A2′) hold. Then the following two statements are valid.(i)
Ifb′(0)<μ, then the zero solution of (52) is globally asymptotically stable in 𝕐+.(ii)
Ifb′(0)>μ, then the zero solution of (52) is unstable, and (52) admits at least one spatially homogeneous steady state w*∈(0,M].Theorem 12.
Assume that (A1) and (A2′) hold, andb′(0)>μ. Then the following two statements for the positive and spatially homogeneous steady state w* of (52) are valid. (i)
Ifb′(w*)>μ, then w* is unstable.(ii)
If-μ≤b′(w*)<μ and (A3′) hold, then w* is globally asymptotically stable in 𝕐+∖{0}.Remark 13.
It is easy to see that (52) is discussed in [10] and some partial results of Theorems 11 and 12 have been obtained [10].Example 2.
Consider the following Nicholson’s blowfly equation resulting from lettingf(w)=μw and b(w)=pwe-qw in (1):
(53)∂w(t,x)∂t=dΔw(t,x)-μw(t,x)+∫Ω‍pw(t-τ,y)e-qw(t-τ,y)k(α,x,y)dy,∂w(t,x)∂n=0,t>0,x∈∂Ω,w(t,x)=ϕ(t,x)≥0,t∈[-τ,0],x∈Ω,
where p and q are two positive constants.By the same arguments as in [10, Section 4], together with Theorems 11 and 12, we have the following results for (53).Theorem 14.
(
i
) If p<μ, then the zero solution of (53) is globally asymptotically stable in 𝕐+.
(
i
i
) If p>μ, then the zero solution of (53) is unstable, and (53) admits the unique positive constant equilibrium w*=(ln⁡(p/μ))/q.Theorem 15.
Ifμ<p≤e2μ, the unique positive constant equilibrium w*=(ln⁡(p/μ))/q of (53) is globally asymptotically stable in 𝕐+∖{0}.Remark 16.
It is easy to see that the equation in [7] is a special case of α=0 of (53). Hence all main results of [7] are special cases of our Theorems 14 and 15.Example 3.
Consider the following Mackey-Glass equation resulting from lettingf(w)=μwl+1 and b(w)=pw/(q+wl) in (1):
(54)∂w(t,x)∂t=dΔw(t,x)-μwl+1(t,x)+∫Ω‍pw(t-τ,y)q+wl(t-τ,y)k(α,x,y)dy,∂w(t,x)∂n=0,t>0,x∈∂Ω,w(t,x)=ϕ(t,x)≥0,t∈[-τ,0],x∈Ω,
where l is a positive constant.By the same arguments as in [10, Section 4], together with Theorems 9 and 10, we have the following results for (54).Theorem 17.
The zero solution of (54) is always unstable, and (54) must admit the unique positive constant equilibrium w*=(Z0)1/l, where
(55)Z0=12(-q+q2+4pμ).Theorem 18.
If((pq+p(1-l)Z0)/(q+Z0)2)>μ(1+l)Z0, the unique positive constant equilibrium w*=(Z0)1/l is unstable, and if -μ(1+l)Z0≤((pq+p(1-l)Z0)/(q+Z0)2)<μ(1+l)Z0 it is globally asymptotically stable in 𝕐+∖{0}.Example 4.
Consider the equation resulting from lettingf(w)=μw2 and b(w)=pw(1-(w/r)) in (1); that is,
(56)∂w(t,x)∂t=dΔw(t,x)-μw2(t,x)+∫Ω‍pw(t-τ,y)(1-w(t-τ,y)r)k(α,x,y)dy,∂w(t,x)∂n=0,t>0,x∈∂Ω,w(t,x)=ϕ(t,x)≥0,t∈[-τ,0],x∈Ω,
where 0<r≤+∞.Clearly,w*=(p/(μ+pr-1)), max⁡w≥0b(w)=b(r/2), and (A1)–(A3) hold, where r-1=0 if r=+∞. Moreover, 0<w*≤r/2 if μ≥pr-1. Therefore, (A4) is satisfied if μ≥pr-1. Thus, Theorems 9 and 10 imply the following results.Theorem 19.
The zero solution of (56) is always unstable, and (56) must admit the unique positive constant equilibrium w*=(p/(μ+pr-1)).Theorem 20.
Ifμ≥pr-1, then the unique positive constant equilibrium w*=(p/(μ+pr-1)) is globally asymptotically stable in 𝕐+∖{0}.Remark 21.
It is easy to see that the equation in [11] is a special case of r=+∞ of (56) and hence some partial results of Theorems 19 and 20 have been obtained [11].



---

*Source: 378172-2014-04-29.xml*