Existence and stability of steady-state solutions of reaction
Transcript Of Existence and stability of steady-state solutions of reaction
Z. Angew. Math. Phys.
(2021) 72:43
c 2021 The Author(s), under exclusive licence to Springer Nature
Switzerland AG part of Springer Nature
https://doi.org/10.1007/s00033-021-01474-1
Zeitschrift fu¨r angewandte Mathematik und Physik ZAMP
Existence and stability of steady-state solutions of reaction–diffusion equations with nonlocal delay effect
Wenjie Zuo and Junping Shi
Abstract. A general reaction–diffusion equation with spatiotemporal delay and homogeneous Dirichlet boundary condition is considered. The existence and stability of positive steady-state solutions are proved via studying an equivalent reaction– diffusion system without nonlocal and delay structure and applying local and global bifurcation theory. The global structure of the set of steady states is characterized according to type of nonlinearities and diffusion coefficient. Our general results are applied to diffusive logistic growth models and Nicholson’s blowflies-type models.
Mathematics Subject Classification. 35K57, 35B32, 35K58, 35Q92, 92D25.
Keywords. Reaction–diffusion equation, Spatiotemporal delay, Dirichlet boundary condition, Stability, Global bifurcation.
1. Introduction
Reaction–diffusion models have been used to describe the evolution of population density in biological or
chemical problems, and the qualitative behavior of solutions to the models can be used to predict outcomes
of natural or engineered biochemical events. Typical long-term behavior of the models is the convergence
to steady-state solutions or time-periodic orbits, or formation of some particular spatiotemporal patterns.
The reaction dynamics of the models often depends on the system states of past time, which induces time
delays in the model equations. Realistic time delay terms in the model distribute over all past time, and
due to the spatial structure and the diffusive nature of population, the time delay is also nonlocal over
the space.
In this paper, we consider a general reaction–diffusion model with spatiotemporal nonlocal delay effect
and Dirichlet boundary conditions:
⎧ ⎪⎨ut(x, t) = dΔu(x, t) + F (λ, u(x, t), (g ∗ ∗H(u))(x, t)), ⎪⎩u(x, t) = 0,
u(x, t) = η(x, t),
x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω, t ∈ (−∞, 0],
(1.1)
where u(x, t) is the population density at time t and location x ∈ Ω ⊂ Rn, d > 0 is the diffusion coefficient, and the initial condition is assumed to be given for all past time; F (λ, u, v) is a nonlinear function depending on a parameter λ, the local population density u(x, t), and a variable v(x, t) representing past
Partially supported by the NSFC of China (No. 11671236), the Natural Science Foundation of Shandong Province of China (No. ZR2019MA006), the Fundamental Research Funds for the Central Universities (No. 19CX02055A), China Scholarship Council and US-NSF grants DMS-1715651 and DMS-1853598.
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state of population density. Here, the past state of population density v(x, t) is given by a form
t
v(x, t) = (g ∗ ∗H(u))(x, t) =
G(x, y, t − s)g(t − s)H(u(y, s))dyds,
(1.2)
−∞ Ω
where the spatial weighing function G(x, y, t − s) means the probability that an individual in location y
moves to location x at a past time t−s, the temporal weighing function g(t−s) characterizes the weight of
past time t − s in the entire past, and H is a function of the state variable u. Here, G : Ω × Ω × (0, ∞) → R is a (generalized) function or measure and g : [0, ∞) → R+ is a probability distribution function satisfying
∞
G(x, y, t)dy = 1, x ∈ Ω, t > 0, and g(t)dt = 1.
(1.3)
Ω
0
The nonlocal distributed delay term g ∗ ∗H(u) is a spatiotemporal average of the past state of density function u. Such nonlocal delay effect was first introduced in [4] when Ω = Rn, and in [18] when Ω is a
bounded domain. See [17,19,40] for more detailed explanation of the nonlocal delay in the population
models.
In this paper, we assume that G(x, y, t) is the Green’s function of diffusion equation with Dirichlet
boundary condition:
∞
G(x, y, t) = e−dλntφn(x)φn(y),
(1.4)
n=1
where λn is the n-th eigenvalue of the following eigenvalue problem
−Δφ(x) = λφ(x), x ∈ Ω,
φ(x) = 0,
x ∈ ∂Ω,
such that
0 < λ1 ≤ λ2 ≤ · · · ≤ λn ≤ · · · → +∞, as n → ∞,
and φn(x) is the corresponding eigenfunction of λn normalized so that (1.3) is satisfied. This assumption is consistent with the diffusive behavior of the population in the past time. On the other hand, the temporal distribution function is chosen to be
gw(t) = τ1 e− τt , gs(t) = τt2 e− τt ,
(1.5)
which are referred as weak kernel and strong kernel. When G and g take the forms in (1.4) and (1.5), the
model (1.1) is equivalent to a system of reaction–diffusion equations without nonlocal and delay effect
(the precise equivalence is described in Sect. 2). For example, when the weak kernel is used, the new
equivalent system is ⎧ ⎪⎪⎨ut(x, t) = dΔu(x, t) + F (λ, u(x, t), v(x, t)), 1 ⎪⎪⎩vt(x, t) = dΔv(x, t) + τ (H(u(x, t)) − v(x, t)), u(x, t) = v(x, t) = 0,
x ∈ Ω, t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0.
(1.6)
We use established techniques for classical reaction–diffusion systems such as local and global bifurcation theory, linear stability analysis, nonlinear elliptic equations, and a priori estimates to study (1.6), which in turn provides information on steady-state solutions and dynamical behavior of reaction–diffusion equation with nonlocal delay effect (1.1). Our results assume general form of the nonlinear functions F and H, hence they can be applied to a wide variety of population growth models in the literature. In particular, we demonstrate our result by applying them to logistic-type models [4], and Nicholson’s blowflies-type models [40].
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Our results can be compared to a vast body of previous work on (1.1) with other choices of G and g as well as other boundary conditions. The spatiotemporal kernel G can take the form: (A) δ(x − y) (local);
(B) K(x, y) (spatial); or (C) the one in (1.4) (diffusion). Special examples of (B) include: (B1) Green’s function of stationary diffusion operator −dΔ + μ; or (B2) constant function. The delay distribution
tne−t/τ function g can take the form: (a) δ(t − τ ) (discrete delay); or (b) gn(t) = τ n+1Γ(n + 1) (Gamma function of order n). Note that gw and gs defined in (1.5) are the Gamma function of order 0 and 1. Finally, the boundary conditions can be: (α) Dirichlet u = 0; (β) Neumann ∂u = 0; or (γ) periodic on Rn. Various
∂n combinations of G, g and boundary conditions have been used for (1.1), and Table 1 gives a partial list of
references which consider (1.1) with these different choices of kernel functions and boundary conditions.
When the spatiotemporal kernel G is a delta function δ(x−y) as type (A), the system (1.1) is spatially local. For discrete-type delay (a), it has been shown that for Neumann boundary value problem, the positive steady-state solution loses its stability via a Hopf bifurcation when the delay τ is large [27,34,47], while the same phenomenon is also proved for small amplitude positive steady state for Dirichlet boundary value problem [5,37,38,42]. A temporally oscillatory solution emerges from the Hopf bifurcation, and this solution is spatially nonhomogeneous under Dirichlet boundary condition [5,37,38,42] or with spatial heterogeneity [34]. Similar Hopf bifurcation and temporally oscillatory solution are also found when the delay is distributed one as type (b) [16,33,49]. When the kernel function G is a spatial one as type (B), the system (1.1) is a nonlocal one. For discrete delay (a) and Dirichlet boundary condition, Hopf bifurcation and spatially nonhomogeneous oscillatory solution bifurcating from small amplitude positive steady state have also been found [8,10,21,22]. The rigorous proof of Hopf bifurcation and spatially nonhomogeneous oscillatory solution bifurcating from large amplitude positive steady state remains an open question, although numerically it has been found in many cases.
For the diffusion kernel defined in (1.4) (C) and Gamma distribution function (b), it is found under Dirichlet boundary condition that the small amplitude positive steady state does not undergo Hopf bifurcation and it remains stable for τ > 0 [9]. Same result holds for Neumann boundary condition and weak kernel, but Hopf bifurcation occurs for Neumann boundary condition and strong kernel [50]. This paper also considers the Dirichlet diffusion kernel defined in (1.4) (C) and weak kernel, and we show that for fixed τ > 0, the bifurcating positive steady-state solution is usually locally asymptotically stable for d ∈ (d∗(τ ) − (τ ), d∗(τ )), where d∗(τ ) is the bifurcation point and (τ ) is a small constant depending on τ . So our results here again confirm the nonoccurrence of Hopf bifurcation for the diffusion kernel case and weak distribution kernel as indicated in [9,50]. The results in this paper take an entirely different approach based on the equivalent system (1.6) and theory of semilinear elliptic systems, and it also holds for much general setting compared to the ones in [9,50]. Some of our existence, stability and uniqueness results are of global nature (see Sects. 5 and 6).
Equation (1.1) has also been used to model biological invasion or spreading behavior, and traveling wave solutions of (1.1) with various choices of G and g have been considered in, for example, [1,2,15,26, 35, 40, 41].
The rest of this paper is organized as follows. In Sect. 2, we prove the equivalence of the system (1.1) with spatiotemporal delay and a system without nonlocal and delay effect. Section 3 is devoted to obtain the existence of the local bifurcated spatially nonhomogeneous steady-state solutions, and the stability of bifurcating solutions is shown in Sect. 4. In Sect. 5, the global bifurcation structure of positive steady-state solutions is shown in two different scenarios, and a uniqueness of positive steady-state result for one-dimensional case is shown in Sect. 6. In Sect. 7, we apply our main results to the logistic-type models and Nicholson’s blowflies-type equations.
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Table 1. References on dynamics of (1.1) with different combinations of G, g and boundary conditions
(α)
(a)
(b)
(β)
(a)
(b)
(γ)
(a)
(b)
(A)
[5, 20, 36–38, 42, 46]
[23, 29, 33]
(A)
[27, 34, 43–45, 47]
[14, 16, 49]
(A)
(B)
[8, 10, 21, 22, 46]
(B)
[28]
(B)
[3]
(C)
[9, 18]
(C)
[18, 39, 50]
(C)
[4]
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2. Equivalence of systems
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In this section, we establish the equivalence of the reaction–diffusion system (1.1) with spatiotemporal
delay given in (1.4) and (1.5) and reaction–diffusion systems without delays. We will consider the cases of bounded domains and entire space Rn.
2.1. The bounded domain
First, we recall the following standard result for the linear parabolic equations.
Lemma 2.1. Let Ω be a bounded domain in Rn with smooth boundary. Suppose that f : Ω × (t0, +∞) is continuous and u ∈ C2,1(Ω × [t0, +∞)) ∩ C0(Ω¯ × [t0, +∞)) satisfies
⎧ ⎪⎨ut(x, t) = dΔu(x, t) − ku(x, t) + f (x, t), ⎪⎩Bu(x, t) = 0,
u(x, t0) = u0(x),
x ∈ Ω, t > t0, x ∈ ∂Ω, t ≥ t0, x ∈ Ω,
(2.1)
where Bu = u, or Bu = ∂u + a(x)u with a(x) ≥ 0. Then, ∂n
t
u(x, t) = G(x, y, t − t0)e−k(t−t0)u0(y)dy +
G(x, y, t − s)e−k(t−s)f (y, s)dyds,
Ω
t0 Ω
where for any fixed y ∈ Ω, G(x, y, t) is the Green function of the diffusion equation satisfying
⎧ ⎪⎨Gt(x, y, t) = dΔxG(x, y, t), ⎪⎩BG(Gx(,xy,,y0,)t=) =δ(0x, − y).
x ∈ Ω, t > 0 x ∈ ∂Ω, t > 0,
(2.2)
Proof. Denote by {(μn, ϕn(x))}∞ n=1 the eigenvalues and the corresponding normalized eigenfunctions of
−Δϕ(x) = μϕ(x), x ∈ Ω,
Bϕ(x) = 0,
x ∈ ∂Ω.
The for the homogeneous equation
⎧ ⎪⎨vt(x, t) = dΔv(x, t) − kv(x, t), ⎪⎩Bv(x, t) = 0,
v(x, t0) = v0(x),
x ∈ Ω, t > t0, x ∈ ∂Ω, t ≥ t0, x ∈ Ω,
the solution is given by
∞
v(x, t) = cne−(dμn+k)(t−t0)ϕn(x),
n=1
cn =
Ω
φn(y)v0(y)dy.
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This implies that
⎛
∞
v(x, t) = ⎝
⎞ ϕn(y)v0(y)dy⎠ e−(dμn+k)(t−t0)ϕn(x)
n=1 Ω
∞
=
e−dμn(t−t0)ϕn(x)ϕn(y) e−k(t−t0)v0(y)dy
Ω n=1
= G(x, y, t − t0)e−k(t−t0)v0(y)dy.
Ω
By the Duhamel principle, it follows that the solution of the initial boundary value problem (2.1) is given by (2.2).
Now we have the following result regarding an entire solution u(x, t) defined for t ∈ (−∞, +∞):
Lemma 2.2. Let Ω be a bounded domain in Rn with smooth boundary. Suppose that f : Ω × (−∞, +∞) is continuous and u ∈ C2,1(Ω × (−∞, +∞)) ∩ C0(Ω × (−∞, +∞)) satisfies
ut(x, t) = dΔu(x, t) − ku(x, t) + f (x, t), x ∈ Ω, t ∈ (−∞, +∞),
Bu(x, t) = 0,
x ∈ ∂Ω, t ∈ (−∞, +∞).
Then,
t
u(x, t) =
G(x, y, t − s)e−k(t−s)f (y, s)dyds.
−∞ Ω
Proof. For any fixed t0 < t, by Lemma 2.1, we have
t
u(x, t) = h(x, t; t0) +
G(x, y, t − s)e−k(t−s)f (y, s)dyds,
t0 Ω
(2.3)
where h(x, t; t0)
G(x, y, t − t0)e−k(t−t0)u(y, t0)dy. And
Ω
h(x, t; t0) ≤ u(·, t0) G(x, y, t − t0)dye−k(t−t0) ≤ u(·, t0) e−k(t−t0).
Ω
Then, h(x, t; t0) → 0 as t0 → −∞ and from the arbitrariness of t0, we let t0 → −∞ and we obtain (2.3).
By using Lemma 2.2, we have the following results on the equivalence of the two systems under the weak or strong distribution kernels.
Proposition 2.3. Suppose that the distributed delay kernel g(t) is given by the weak kernel function gw(t) =
1
e−
t τ
,
and
define
τ
t
v(x, t) = (gw ∗ ∗H(u))(x, t) =
G(x, y, t − s)gw(t − s)H(u(y, s))dyds.
−∞ Ω
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1. If u(x, t) is the solution of (1.1), then (u(x, t), v(x, t)) is the solution of
⎧
⎪⎪⎪⎪⎪ut(x,
t)
=
dΔu(x,
t)
+
F 1
(λ,
u(x,
t),
v(x,
t)),
⎪⎪⎪⎪⎪⎨Bvtu(x(x, t,)t)==dΔBvv((xx,, tt)) += 0τ,(H(u(x, t)) − v(x, t)),
x ∈ Ω, t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0,
⎪⎪⎪⎪⎪u(x,
0)
=
η(x, 0),
0
⎪⎪⎪⎪⎪⎩v(x, 0) = τ1
G(x,
y
,
−s)e
s τ
H
(η
(y
,
s))dy
ds,
−∞ Ω
x ∈ Ω, x ∈ Ω.
(2.4)
2. If (u(x, t), v(x, t)) is a solution of ⎧ ⎪⎪⎨ut(x, t) = dΔu(x, t) + F (λ, u(x, t), v(x, t)), 1 ⎪⎪⎩vt(x, t) = dΔv(x, t) + τ (H(u(x, t)) − v(x, t)), Bu(x, t) = Bv(x, t) = 0,
x ∈ Ω, t ∈ R, x ∈ Ω, t ∈ R, x ∈ ∂Ω, t ∈ R.
(2.5)
Then, u(x, t) satisfies (1.1) such that η(x, s) = u(x, s), −∞ < s < 0. In particular, if (u(x), v(x)) is a steady-state solution of (2.4), then u(x) is a steady-state solution of (1.1); and if (u(x, t), v(x, t)) is a periodic solution of (2.5) with period T , then u(x, t) is a periodic solution of (1.1) with period T.
Proposition 2.4. Suppose that the distributed delay kernel g(t) is given by the strong kernel function gs(t) = τt2 e− τt , and define
t
v(x, t) = (gs ∗ ∗H(u))(x, t) =
G(x, y, t − s)gs(t − s)H(u(y, s))dyds.
(2.6)
−∞ Ω
1. If u(x, t) is the solution of (1.1), then (u(x, t), v(x, t), w(x, t)) is the solution of
⎧
⎪⎪⎪⎪⎪ut(x,
t)
=
dΔu(x,
t)
+
F 1
(λ,
u(x,
t),
v(x,
t)),
⎪⎪⎪⎪⎪vt(x, t) = dΔv(x, t) + τ (w(x, t) − v(x, t)),
⎪⎪⎪⎪⎪wt(x, t)
=
dΔw(x, t)
+
1 (H(u(x, t))
−
w(x, t)),
⎪⎪⎪⎪⎪⎨Bu(x, t) = Bv(x, t) = Bτw(x, t) = 0,
u(x, 0) = η(x, 0),
⎪⎪⎪⎪⎪
0
−s s
⎪⎪⎪⎪⎪v(x, 0) =
G(x, y, −s) τ 2 e τ H(η(y, s))dyds,
⎪⎪⎪⎪⎪
−∞ Ω 0
⎪⎪⎪⎪⎪⎩w(x, 0) =
G(x, y, −s) 1 e τs H(η(y, s))dyds, τ
x ∈ Ω, t > 0, x ∈ Ω, t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,
x ∈ Ω,
x ∈ Ω.
−∞ Ω
(2.7)
2. If (u(x, t), v(x, t), w(x, t)) is a solution of ⎧ ⎪⎪⎪⎪⎪⎨vutt((xx,,tt)) == ddΔΔvu((xx,,tt))++ F1 ((wλ,(xu,(xt), −t),vv((xx,,tt)))),, τ ⎪⎪⎪⎪⎪⎩wt(x, t) = dΔw(x, t) + τ1 (H(u(x, t)) − w(x, t)), Bu(x, t) = Bv(x, t) = Bw(x, t) = 0,
x ∈ Ω, t ∈ R, x ∈ Ω, t ∈ R, x ∈ Ω, t ∈ R, x ∈ ∂Ω, t ∈ R.
(2.8)
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Then, u(x, t) satisfies (1.1) with the strong kernel gs(t) such that η(x, s) = u(x, s), −∞ < s < 0. In particular, if (u(x), v(x), w(x)) is a steady-state solution of (2.7), then u(x) is a steady-state solution of (1.1); if (u(x, t), v(x, t), w(x, t)) is a periodic solution of (2.8) with period T , then u(x, t) is a periodic solution of (1.1) with period T .
The proof of Proposition 2.3 is immediate from Lemma 2.2, and the proof of Proposition 2.4 follows from differentiating (2.6) with respect to t and elementary calculation. The equivalence of (1.1) and (2.7) has been first observed in [18].
2.2. The whole space RN
Consider a general scalar reaction–diffusion equation with spatiotemporal delay in the entire space:
ut(x, t) = dΔu(x, t) + F (λ, u(x, t), (g ∗ ∗H(u))(x, t)), x ∈ RN , t ∈ R.
(2.9)
Here,
t
(g ∗ ∗H(u))(x, t) =
G(x, y, t − s)g(t − s)H(u(y, s))dyds,
−∞ RN
where for y ∈ RN , G(x, y, t) is a fundamental solution of
Gt(x, y, t) = dΔxG(x, y, t), x ∈ RN , t > 0,
G(x, y, 0) = δ(x − y),
x ∈ RN , t > 0.
By using the similar method as Propositions 2.3 and 2.4, we can prove the following results on equivalence of (2.9) and associated systems:
Proposition 2.5.
1. If (u(x, t), v(x, t)) is a solution of
⎧ ⎨ut(x, t) = dΔu(x, t) + F (λ, u, v),
⎩vt(x, t)
=
dΔv(x, t)
+
1 (H(u(x, t))
−
v(x, t)),
τ
x ∈ RN , t ∈ R, x ∈ RN , t ∈ R,
then
u(x, t)
is
also
a
solution
of
(2.9)
with
the
weak
kernel
gw (t)
=
1
e−
t τ
.
τ
2. If (u(x, t), v(x, t), w(x, t)) is a solution of
⎧
⎪⎪⎪⎨ut(x,
t)
=
dΔu(x,
t)
+
F 1
(λ,
u,
v),
vt(x, t) = dΔv(x, t) + (w(x, t) − v(x, t)),
⎪⎪⎪⎩wt(x, t) = dΔw(x, t) +ττ1 (H(u(x, t)) − w(x, t)),
x ∈ RN , t ∈ R, x ∈ RN , t ∈ R, x ∈ RN , t ∈ R,
then u(x, t) is also a solution of (2.9) with the strong kernel gs(t) = τt2 e− τt .
Note that the equivalence of systems is valid for any solution defined for all t ∈ R, which include steady-state solutions, periodic solutions, and also traveling wave solutions. This equivalence was first observed in [4]. In this paper, we only consider the bounded domain case.
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3. Existence and local bifurcation of steady-state solutions
In this section, we consider the existence of positive steady-state solution of the system (1.1) with a weak
kernel subject to Dirichlet boundary condition. The strong kernel case can be considered similarly but
will not be considered here. By Theorem 2.3, we only need to consider the steady-state solutions of the
equivalent system (2.4), which are the solutions of system of semilinear elliptic system:
⎧ ⎪⎪⎨dΔu(x) + F (λ, u(x), v(x)) = 0,
1 ⎪⎪⎩dΔv(x) + τ (H(u(x)) − v(x)) = 0,
u(x) = v(x) = 0,
x ∈ Ω, x ∈ Ω, x ∈ ∂Ω.
(3.1)
In the following, we always assume that d > 0, τ > 0 and λ ≥ 0. We use bifurcation method with
parameter d to prove the existence of positive solutions to (3.1). Note that a bifurcation analysis can also
be conducted using parameter λ with a fixed d. So in the following, we assume F (λ, u, v) ≡ F (u, v) as λ
is fixed, so we consider
⎧ ⎪⎪⎨dΔu(x) + F (u(x), v(x)) = 0,
1 ⎪⎪⎩dΔv(x) + τ (H(u(x)) − v(x)) = 0,
u(x) = v(x) = 0,
x ∈ Ω, x ∈ Ω, x ∈ ∂Ω.
(3.2)
We assume that the nonlinearities F (u, v) and H(u) in (3.2) satisfy
(A1) There exists a δ > 0 such that F : Uδ × Uδ → R and H : Uδ → R are C2 functions, where Uδ = {y ∈ R : |y| < δ};
(A2) F (0, 0) = 0, H(0) = 0 and H (0) > 0.
In the following, the first and second derivatives of F and H are denoted by
Fu(0, 0) = a, Fv(0, 0) = b, H (0) = k > 0, Fuu(0, 0) = p, Fuv(0, 0) = q, Fvv(0, 0) = r, H (0) = l.
(3.3)
From (A2), it is known that (u, v) = (0, 0) is a trivial solution of (3.2) for any d, τ > 0. Let X = W 2,p(Ω) × W01,p(Ω) for p > n, and let Y = Lp(Ω). For the bifurcation of positive solutions of (3.2), fixing τ > 0, we define a nonlinear mapping W : R × X2 → Y 2 by
dΔu + F (u, v) W (d, u, v) = dΔv + 1 (H(u) − v) .
τ
Then, a solution (d, u, v) of (3.2) is equivalent to W (d, u, v) = (0, 0)T . Our main result on the local bifurcation of positive solutions of (3.2) is as follows:
(3.4)
Theorem 3.1. Suppose that τ > 0 is fixed, the conditions (A1) and (A2) hold, and also
(A3) a + bk > 0.
Define
d∗(τ ) = 1 (aτ − 1 + 2λ1τ
(aτ + 1)2 + 4bτ k),
(3.5)
where λ1 is the principal eigenvalue of −Δ in H01(Ω) with corresponding eigenfunction φ1(x) > 0. Then,
1. d = d∗ = d∗(τ ) is the unique bifurcation point of the system (3.2) where positive solutions of (3.2)
bifurcate from the line of trivial solutions Γ0 = {(d, 0, 0) : d > 0}.
43 Page 10 of 26
W. Zuo and J. Shi
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2. Near (d, u, v) = (d∗, 0, 0), there exists δ1 > 0 such that all positive solutions of (3.2) near the bifurcation point lie on a smooth curve Γ1 = {(d(s), ud(s, ·), vd(s, ·)) : s ∈ (0, δ1)} with d(s) = d∗ + d (0)s + s2z0(s), (ud(s, ·), vd(s, ·)) = s(1, M )φ1(·) + s2(z1(s, ·), z2(s, ·)), where
M= aτ + 1 +
2k ,
(aτ + 1)2 + 4bτ k
(3.6)
such that z0 : (0, δ1) → R and z1, z2 : (0, δ1) → X are smooth functions satisfying zi(0) = 0 for i = 0, 1, 2. Moreover,
[k(p + 2qM + rM 2) + bM l] φ31(x)dx
d (0) =
Ω
.
2λ1(k + M 2bτ ) φ21(x)dx
Ω
(3.7)
Proof. Let W be defined as in (3.4). Then, from (A1), W is twice differentiable in R × Xδ2, where Xδ is an open neighborhood of 0 in X. The Fr´echet derivative of W in variable (u, v) is
ξ1
dΔξ1 + Fu(u, v)ξ1 + Fv(u, v)ξ2
W(u,v)(d, u, v) ξ2 =
dΔξ2 + 1 (H (u)ξ1 − ξ2)
,
τ
and in particular when (u, v) = (0, 0),
(3.8)
W(u,v)(d, 0, 0) ξξ12 = ddΔΔξξ12 + A ξξ12 ,
(3.9)
where A is defined by
ab A = k −1 .
ττ The eigenvalues of A satisfy the characteristic equation
(3.10)
μ2 − a − 1 μ − a + bk = 0.
τ
τ
From (A3), we have a + bk > 0, then it is easy to see that A has a unique positive eigenvalue μ1 > 0
defined by
μ1 = 1 (aτ − 1 + (aτ + 1)2 + 4bτ k) 2τ
(3.11)
with a positive eigenvector (1, M ) where M is defined in (3.6). From the implicit function theorem, if d > 0
is a bifurcation point for positive solutions of (3.2) from the line of trivial solutions, then W(u,v)(d, 0, 0) is not invertible. That is, the null space N (W(u,v)(d, 0, 0)) = {0}. From Fourier theory, we must have d = μ1/λn, where λn is an eigenvalue of −Δ in H01(Ω). Since φ1 is the only eigenfunction which does not change sign in Ω, the only possible bifurcation point for positive solutions is d = d∗ = μ1/λ1 which is
given by (3.5). At (d∗, 0, 0), it is easy to compute the kernels of the linearized operator W(u,v)(d∗, 0, 0) and associated
adjoint operator W(∗u,v)(d∗, 0, 0), respectively:
N (W(u,v)(d∗, 0, 0)) = span{(1, M )φ1}, N (W(∗u,v)(d∗, 0, 0)) = span{(1, M bτ /k)φ1}.
And the range of the operator W(u,v)(d∗, 0, 0) is described by the following form:
⎧
⎫
⎨
⎬
R(W(u,v)(d∗, 0, 0)) = ⎩(g1, g2) ∈ Y 2 : (kg1(x) + M bτ g2(x)) φ1(x)dx = 0⎭ .
Ω
(2021) 72:43
c 2021 The Author(s), under exclusive licence to Springer Nature
Switzerland AG part of Springer Nature
https://doi.org/10.1007/s00033-021-01474-1
Zeitschrift fu¨r angewandte Mathematik und Physik ZAMP
Existence and stability of steady-state solutions of reaction–diffusion equations with nonlocal delay effect
Wenjie Zuo and Junping Shi
Abstract. A general reaction–diffusion equation with spatiotemporal delay and homogeneous Dirichlet boundary condition is considered. The existence and stability of positive steady-state solutions are proved via studying an equivalent reaction– diffusion system without nonlocal and delay structure and applying local and global bifurcation theory. The global structure of the set of steady states is characterized according to type of nonlinearities and diffusion coefficient. Our general results are applied to diffusive logistic growth models and Nicholson’s blowflies-type models.
Mathematics Subject Classification. 35K57, 35B32, 35K58, 35Q92, 92D25.
Keywords. Reaction–diffusion equation, Spatiotemporal delay, Dirichlet boundary condition, Stability, Global bifurcation.
1. Introduction
Reaction–diffusion models have been used to describe the evolution of population density in biological or
chemical problems, and the qualitative behavior of solutions to the models can be used to predict outcomes
of natural or engineered biochemical events. Typical long-term behavior of the models is the convergence
to steady-state solutions or time-periodic orbits, or formation of some particular spatiotemporal patterns.
The reaction dynamics of the models often depends on the system states of past time, which induces time
delays in the model equations. Realistic time delay terms in the model distribute over all past time, and
due to the spatial structure and the diffusive nature of population, the time delay is also nonlocal over
the space.
In this paper, we consider a general reaction–diffusion model with spatiotemporal nonlocal delay effect
and Dirichlet boundary conditions:
⎧ ⎪⎨ut(x, t) = dΔu(x, t) + F (λ, u(x, t), (g ∗ ∗H(u))(x, t)), ⎪⎩u(x, t) = 0,
u(x, t) = η(x, t),
x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω, t ∈ (−∞, 0],
(1.1)
where u(x, t) is the population density at time t and location x ∈ Ω ⊂ Rn, d > 0 is the diffusion coefficient, and the initial condition is assumed to be given for all past time; F (λ, u, v) is a nonlinear function depending on a parameter λ, the local population density u(x, t), and a variable v(x, t) representing past
Partially supported by the NSFC of China (No. 11671236), the Natural Science Foundation of Shandong Province of China (No. ZR2019MA006), the Fundamental Research Funds for the Central Universities (No. 19CX02055A), China Scholarship Council and US-NSF grants DMS-1715651 and DMS-1853598.
0123456789().: V,-vol
43 Page 2 of 26
W. Zuo and J. Shi
ZAMP
state of population density. Here, the past state of population density v(x, t) is given by a form
t
v(x, t) = (g ∗ ∗H(u))(x, t) =
G(x, y, t − s)g(t − s)H(u(y, s))dyds,
(1.2)
−∞ Ω
where the spatial weighing function G(x, y, t − s) means the probability that an individual in location y
moves to location x at a past time t−s, the temporal weighing function g(t−s) characterizes the weight of
past time t − s in the entire past, and H is a function of the state variable u. Here, G : Ω × Ω × (0, ∞) → R is a (generalized) function or measure and g : [0, ∞) → R+ is a probability distribution function satisfying
∞
G(x, y, t)dy = 1, x ∈ Ω, t > 0, and g(t)dt = 1.
(1.3)
Ω
0
The nonlocal distributed delay term g ∗ ∗H(u) is a spatiotemporal average of the past state of density function u. Such nonlocal delay effect was first introduced in [4] when Ω = Rn, and in [18] when Ω is a
bounded domain. See [17,19,40] for more detailed explanation of the nonlocal delay in the population
models.
In this paper, we assume that G(x, y, t) is the Green’s function of diffusion equation with Dirichlet
boundary condition:
∞
G(x, y, t) = e−dλntφn(x)φn(y),
(1.4)
n=1
where λn is the n-th eigenvalue of the following eigenvalue problem
−Δφ(x) = λφ(x), x ∈ Ω,
φ(x) = 0,
x ∈ ∂Ω,
such that
0 < λ1 ≤ λ2 ≤ · · · ≤ λn ≤ · · · → +∞, as n → ∞,
and φn(x) is the corresponding eigenfunction of λn normalized so that (1.3) is satisfied. This assumption is consistent with the diffusive behavior of the population in the past time. On the other hand, the temporal distribution function is chosen to be
gw(t) = τ1 e− τt , gs(t) = τt2 e− τt ,
(1.5)
which are referred as weak kernel and strong kernel. When G and g take the forms in (1.4) and (1.5), the
model (1.1) is equivalent to a system of reaction–diffusion equations without nonlocal and delay effect
(the precise equivalence is described in Sect. 2). For example, when the weak kernel is used, the new
equivalent system is ⎧ ⎪⎪⎨ut(x, t) = dΔu(x, t) + F (λ, u(x, t), v(x, t)), 1 ⎪⎪⎩vt(x, t) = dΔv(x, t) + τ (H(u(x, t)) − v(x, t)), u(x, t) = v(x, t) = 0,
x ∈ Ω, t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0.
(1.6)
We use established techniques for classical reaction–diffusion systems such as local and global bifurcation theory, linear stability analysis, nonlinear elliptic equations, and a priori estimates to study (1.6), which in turn provides information on steady-state solutions and dynamical behavior of reaction–diffusion equation with nonlocal delay effect (1.1). Our results assume general form of the nonlinear functions F and H, hence they can be applied to a wide variety of population growth models in the literature. In particular, we demonstrate our result by applying them to logistic-type models [4], and Nicholson’s blowflies-type models [40].
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Existence and stability of steady-state solutions...
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Our results can be compared to a vast body of previous work on (1.1) with other choices of G and g as well as other boundary conditions. The spatiotemporal kernel G can take the form: (A) δ(x − y) (local);
(B) K(x, y) (spatial); or (C) the one in (1.4) (diffusion). Special examples of (B) include: (B1) Green’s function of stationary diffusion operator −dΔ + μ; or (B2) constant function. The delay distribution
tne−t/τ function g can take the form: (a) δ(t − τ ) (discrete delay); or (b) gn(t) = τ n+1Γ(n + 1) (Gamma function of order n). Note that gw and gs defined in (1.5) are the Gamma function of order 0 and 1. Finally, the boundary conditions can be: (α) Dirichlet u = 0; (β) Neumann ∂u = 0; or (γ) periodic on Rn. Various
∂n combinations of G, g and boundary conditions have been used for (1.1), and Table 1 gives a partial list of
references which consider (1.1) with these different choices of kernel functions and boundary conditions.
When the spatiotemporal kernel G is a delta function δ(x−y) as type (A), the system (1.1) is spatially local. For discrete-type delay (a), it has been shown that for Neumann boundary value problem, the positive steady-state solution loses its stability via a Hopf bifurcation when the delay τ is large [27,34,47], while the same phenomenon is also proved for small amplitude positive steady state for Dirichlet boundary value problem [5,37,38,42]. A temporally oscillatory solution emerges from the Hopf bifurcation, and this solution is spatially nonhomogeneous under Dirichlet boundary condition [5,37,38,42] or with spatial heterogeneity [34]. Similar Hopf bifurcation and temporally oscillatory solution are also found when the delay is distributed one as type (b) [16,33,49]. When the kernel function G is a spatial one as type (B), the system (1.1) is a nonlocal one. For discrete delay (a) and Dirichlet boundary condition, Hopf bifurcation and spatially nonhomogeneous oscillatory solution bifurcating from small amplitude positive steady state have also been found [8,10,21,22]. The rigorous proof of Hopf bifurcation and spatially nonhomogeneous oscillatory solution bifurcating from large amplitude positive steady state remains an open question, although numerically it has been found in many cases.
For the diffusion kernel defined in (1.4) (C) and Gamma distribution function (b), it is found under Dirichlet boundary condition that the small amplitude positive steady state does not undergo Hopf bifurcation and it remains stable for τ > 0 [9]. Same result holds for Neumann boundary condition and weak kernel, but Hopf bifurcation occurs for Neumann boundary condition and strong kernel [50]. This paper also considers the Dirichlet diffusion kernel defined in (1.4) (C) and weak kernel, and we show that for fixed τ > 0, the bifurcating positive steady-state solution is usually locally asymptotically stable for d ∈ (d∗(τ ) − (τ ), d∗(τ )), where d∗(τ ) is the bifurcation point and (τ ) is a small constant depending on τ . So our results here again confirm the nonoccurrence of Hopf bifurcation for the diffusion kernel case and weak distribution kernel as indicated in [9,50]. The results in this paper take an entirely different approach based on the equivalent system (1.6) and theory of semilinear elliptic systems, and it also holds for much general setting compared to the ones in [9,50]. Some of our existence, stability and uniqueness results are of global nature (see Sects. 5 and 6).
Equation (1.1) has also been used to model biological invasion or spreading behavior, and traveling wave solutions of (1.1) with various choices of G and g have been considered in, for example, [1,2,15,26, 35, 40, 41].
The rest of this paper is organized as follows. In Sect. 2, we prove the equivalence of the system (1.1) with spatiotemporal delay and a system without nonlocal and delay effect. Section 3 is devoted to obtain the existence of the local bifurcated spatially nonhomogeneous steady-state solutions, and the stability of bifurcating solutions is shown in Sect. 4. In Sect. 5, the global bifurcation structure of positive steady-state solutions is shown in two different scenarios, and a uniqueness of positive steady-state result for one-dimensional case is shown in Sect. 6. In Sect. 7, we apply our main results to the logistic-type models and Nicholson’s blowflies-type equations.
43 Page 4 of 26
W. Zuo and J. Shi
Table 1. References on dynamics of (1.1) with different combinations of G, g and boundary conditions
(α)
(a)
(b)
(β)
(a)
(b)
(γ)
(a)
(b)
(A)
[5, 20, 36–38, 42, 46]
[23, 29, 33]
(A)
[27, 34, 43–45, 47]
[14, 16, 49]
(A)
(B)
[8, 10, 21, 22, 46]
(B)
[28]
(B)
[3]
(C)
[9, 18]
(C)
[18, 39, 50]
(C)
[4]
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2. Equivalence of systems
Page 5 of 26 43
In this section, we establish the equivalence of the reaction–diffusion system (1.1) with spatiotemporal
delay given in (1.4) and (1.5) and reaction–diffusion systems without delays. We will consider the cases of bounded domains and entire space Rn.
2.1. The bounded domain
First, we recall the following standard result for the linear parabolic equations.
Lemma 2.1. Let Ω be a bounded domain in Rn with smooth boundary. Suppose that f : Ω × (t0, +∞) is continuous and u ∈ C2,1(Ω × [t0, +∞)) ∩ C0(Ω¯ × [t0, +∞)) satisfies
⎧ ⎪⎨ut(x, t) = dΔu(x, t) − ku(x, t) + f (x, t), ⎪⎩Bu(x, t) = 0,
u(x, t0) = u0(x),
x ∈ Ω, t > t0, x ∈ ∂Ω, t ≥ t0, x ∈ Ω,
(2.1)
where Bu = u, or Bu = ∂u + a(x)u with a(x) ≥ 0. Then, ∂n
t
u(x, t) = G(x, y, t − t0)e−k(t−t0)u0(y)dy +
G(x, y, t − s)e−k(t−s)f (y, s)dyds,
Ω
t0 Ω
where for any fixed y ∈ Ω, G(x, y, t) is the Green function of the diffusion equation satisfying
⎧ ⎪⎨Gt(x, y, t) = dΔxG(x, y, t), ⎪⎩BG(Gx(,xy,,y0,)t=) =δ(0x, − y).
x ∈ Ω, t > 0 x ∈ ∂Ω, t > 0,
(2.2)
Proof. Denote by {(μn, ϕn(x))}∞ n=1 the eigenvalues and the corresponding normalized eigenfunctions of
−Δϕ(x) = μϕ(x), x ∈ Ω,
Bϕ(x) = 0,
x ∈ ∂Ω.
The for the homogeneous equation
⎧ ⎪⎨vt(x, t) = dΔv(x, t) − kv(x, t), ⎪⎩Bv(x, t) = 0,
v(x, t0) = v0(x),
x ∈ Ω, t > t0, x ∈ ∂Ω, t ≥ t0, x ∈ Ω,
the solution is given by
∞
v(x, t) = cne−(dμn+k)(t−t0)ϕn(x),
n=1
cn =
Ω
φn(y)v0(y)dy.
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This implies that
⎛
∞
v(x, t) = ⎝
⎞ ϕn(y)v0(y)dy⎠ e−(dμn+k)(t−t0)ϕn(x)
n=1 Ω
∞
=
e−dμn(t−t0)ϕn(x)ϕn(y) e−k(t−t0)v0(y)dy
Ω n=1
= G(x, y, t − t0)e−k(t−t0)v0(y)dy.
Ω
By the Duhamel principle, it follows that the solution of the initial boundary value problem (2.1) is given by (2.2).
Now we have the following result regarding an entire solution u(x, t) defined for t ∈ (−∞, +∞):
Lemma 2.2. Let Ω be a bounded domain in Rn with smooth boundary. Suppose that f : Ω × (−∞, +∞) is continuous and u ∈ C2,1(Ω × (−∞, +∞)) ∩ C0(Ω × (−∞, +∞)) satisfies
ut(x, t) = dΔu(x, t) − ku(x, t) + f (x, t), x ∈ Ω, t ∈ (−∞, +∞),
Bu(x, t) = 0,
x ∈ ∂Ω, t ∈ (−∞, +∞).
Then,
t
u(x, t) =
G(x, y, t − s)e−k(t−s)f (y, s)dyds.
−∞ Ω
Proof. For any fixed t0 < t, by Lemma 2.1, we have
t
u(x, t) = h(x, t; t0) +
G(x, y, t − s)e−k(t−s)f (y, s)dyds,
t0 Ω
(2.3)
where h(x, t; t0)
G(x, y, t − t0)e−k(t−t0)u(y, t0)dy. And
Ω
h(x, t; t0) ≤ u(·, t0) G(x, y, t − t0)dye−k(t−t0) ≤ u(·, t0) e−k(t−t0).
Ω
Then, h(x, t; t0) → 0 as t0 → −∞ and from the arbitrariness of t0, we let t0 → −∞ and we obtain (2.3).
By using Lemma 2.2, we have the following results on the equivalence of the two systems under the weak or strong distribution kernels.
Proposition 2.3. Suppose that the distributed delay kernel g(t) is given by the weak kernel function gw(t) =
1
e−
t τ
,
and
define
τ
t
v(x, t) = (gw ∗ ∗H(u))(x, t) =
G(x, y, t − s)gw(t − s)H(u(y, s))dyds.
−∞ Ω
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Existence and stability of steady-state solutions...
Page 7 of 26 43
1. If u(x, t) is the solution of (1.1), then (u(x, t), v(x, t)) is the solution of
⎧
⎪⎪⎪⎪⎪ut(x,
t)
=
dΔu(x,
t)
+
F 1
(λ,
u(x,
t),
v(x,
t)),
⎪⎪⎪⎪⎪⎨Bvtu(x(x, t,)t)==dΔBvv((xx,, tt)) += 0τ,(H(u(x, t)) − v(x, t)),
x ∈ Ω, t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0,
⎪⎪⎪⎪⎪u(x,
0)
=
η(x, 0),
0
⎪⎪⎪⎪⎪⎩v(x, 0) = τ1
G(x,
y
,
−s)e
s τ
H
(η
(y
,
s))dy
ds,
−∞ Ω
x ∈ Ω, x ∈ Ω.
(2.4)
2. If (u(x, t), v(x, t)) is a solution of ⎧ ⎪⎪⎨ut(x, t) = dΔu(x, t) + F (λ, u(x, t), v(x, t)), 1 ⎪⎪⎩vt(x, t) = dΔv(x, t) + τ (H(u(x, t)) − v(x, t)), Bu(x, t) = Bv(x, t) = 0,
x ∈ Ω, t ∈ R, x ∈ Ω, t ∈ R, x ∈ ∂Ω, t ∈ R.
(2.5)
Then, u(x, t) satisfies (1.1) such that η(x, s) = u(x, s), −∞ < s < 0. In particular, if (u(x), v(x)) is a steady-state solution of (2.4), then u(x) is a steady-state solution of (1.1); and if (u(x, t), v(x, t)) is a periodic solution of (2.5) with period T , then u(x, t) is a periodic solution of (1.1) with period T.
Proposition 2.4. Suppose that the distributed delay kernel g(t) is given by the strong kernel function gs(t) = τt2 e− τt , and define
t
v(x, t) = (gs ∗ ∗H(u))(x, t) =
G(x, y, t − s)gs(t − s)H(u(y, s))dyds.
(2.6)
−∞ Ω
1. If u(x, t) is the solution of (1.1), then (u(x, t), v(x, t), w(x, t)) is the solution of
⎧
⎪⎪⎪⎪⎪ut(x,
t)
=
dΔu(x,
t)
+
F 1
(λ,
u(x,
t),
v(x,
t)),
⎪⎪⎪⎪⎪vt(x, t) = dΔv(x, t) + τ (w(x, t) − v(x, t)),
⎪⎪⎪⎪⎪wt(x, t)
=
dΔw(x, t)
+
1 (H(u(x, t))
−
w(x, t)),
⎪⎪⎪⎪⎪⎨Bu(x, t) = Bv(x, t) = Bτw(x, t) = 0,
u(x, 0) = η(x, 0),
⎪⎪⎪⎪⎪
0
−s s
⎪⎪⎪⎪⎪v(x, 0) =
G(x, y, −s) τ 2 e τ H(η(y, s))dyds,
⎪⎪⎪⎪⎪
−∞ Ω 0
⎪⎪⎪⎪⎪⎩w(x, 0) =
G(x, y, −s) 1 e τs H(η(y, s))dyds, τ
x ∈ Ω, t > 0, x ∈ Ω, t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω,
x ∈ Ω,
x ∈ Ω.
−∞ Ω
(2.7)
2. If (u(x, t), v(x, t), w(x, t)) is a solution of ⎧ ⎪⎪⎪⎪⎪⎨vutt((xx,,tt)) == ddΔΔvu((xx,,tt))++ F1 ((wλ,(xu,(xt), −t),vv((xx,,tt)))),, τ ⎪⎪⎪⎪⎪⎩wt(x, t) = dΔw(x, t) + τ1 (H(u(x, t)) − w(x, t)), Bu(x, t) = Bv(x, t) = Bw(x, t) = 0,
x ∈ Ω, t ∈ R, x ∈ Ω, t ∈ R, x ∈ Ω, t ∈ R, x ∈ ∂Ω, t ∈ R.
(2.8)
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W. Zuo and J. Shi
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Then, u(x, t) satisfies (1.1) with the strong kernel gs(t) such that η(x, s) = u(x, s), −∞ < s < 0. In particular, if (u(x), v(x), w(x)) is a steady-state solution of (2.7), then u(x) is a steady-state solution of (1.1); if (u(x, t), v(x, t), w(x, t)) is a periodic solution of (2.8) with period T , then u(x, t) is a periodic solution of (1.1) with period T .
The proof of Proposition 2.3 is immediate from Lemma 2.2, and the proof of Proposition 2.4 follows from differentiating (2.6) with respect to t and elementary calculation. The equivalence of (1.1) and (2.7) has been first observed in [18].
2.2. The whole space RN
Consider a general scalar reaction–diffusion equation with spatiotemporal delay in the entire space:
ut(x, t) = dΔu(x, t) + F (λ, u(x, t), (g ∗ ∗H(u))(x, t)), x ∈ RN , t ∈ R.
(2.9)
Here,
t
(g ∗ ∗H(u))(x, t) =
G(x, y, t − s)g(t − s)H(u(y, s))dyds,
−∞ RN
where for y ∈ RN , G(x, y, t) is a fundamental solution of
Gt(x, y, t) = dΔxG(x, y, t), x ∈ RN , t > 0,
G(x, y, 0) = δ(x − y),
x ∈ RN , t > 0.
By using the similar method as Propositions 2.3 and 2.4, we can prove the following results on equivalence of (2.9) and associated systems:
Proposition 2.5.
1. If (u(x, t), v(x, t)) is a solution of
⎧ ⎨ut(x, t) = dΔu(x, t) + F (λ, u, v),
⎩vt(x, t)
=
dΔv(x, t)
+
1 (H(u(x, t))
−
v(x, t)),
τ
x ∈ RN , t ∈ R, x ∈ RN , t ∈ R,
then
u(x, t)
is
also
a
solution
of
(2.9)
with
the
weak
kernel
gw (t)
=
1
e−
t τ
.
τ
2. If (u(x, t), v(x, t), w(x, t)) is a solution of
⎧
⎪⎪⎪⎨ut(x,
t)
=
dΔu(x,
t)
+
F 1
(λ,
u,
v),
vt(x, t) = dΔv(x, t) + (w(x, t) − v(x, t)),
⎪⎪⎪⎩wt(x, t) = dΔw(x, t) +ττ1 (H(u(x, t)) − w(x, t)),
x ∈ RN , t ∈ R, x ∈ RN , t ∈ R, x ∈ RN , t ∈ R,
then u(x, t) is also a solution of (2.9) with the strong kernel gs(t) = τt2 e− τt .
Note that the equivalence of systems is valid for any solution defined for all t ∈ R, which include steady-state solutions, periodic solutions, and also traveling wave solutions. This equivalence was first observed in [4]. In this paper, we only consider the bounded domain case.
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Existence and stability of steady-state solutions...
Page 9 of 26 43
3. Existence and local bifurcation of steady-state solutions
In this section, we consider the existence of positive steady-state solution of the system (1.1) with a weak
kernel subject to Dirichlet boundary condition. The strong kernel case can be considered similarly but
will not be considered here. By Theorem 2.3, we only need to consider the steady-state solutions of the
equivalent system (2.4), which are the solutions of system of semilinear elliptic system:
⎧ ⎪⎪⎨dΔu(x) + F (λ, u(x), v(x)) = 0,
1 ⎪⎪⎩dΔv(x) + τ (H(u(x)) − v(x)) = 0,
u(x) = v(x) = 0,
x ∈ Ω, x ∈ Ω, x ∈ ∂Ω.
(3.1)
In the following, we always assume that d > 0, τ > 0 and λ ≥ 0. We use bifurcation method with
parameter d to prove the existence of positive solutions to (3.1). Note that a bifurcation analysis can also
be conducted using parameter λ with a fixed d. So in the following, we assume F (λ, u, v) ≡ F (u, v) as λ
is fixed, so we consider
⎧ ⎪⎪⎨dΔu(x) + F (u(x), v(x)) = 0,
1 ⎪⎪⎩dΔv(x) + τ (H(u(x)) − v(x)) = 0,
u(x) = v(x) = 0,
x ∈ Ω, x ∈ Ω, x ∈ ∂Ω.
(3.2)
We assume that the nonlinearities F (u, v) and H(u) in (3.2) satisfy
(A1) There exists a δ > 0 such that F : Uδ × Uδ → R and H : Uδ → R are C2 functions, where Uδ = {y ∈ R : |y| < δ};
(A2) F (0, 0) = 0, H(0) = 0 and H (0) > 0.
In the following, the first and second derivatives of F and H are denoted by
Fu(0, 0) = a, Fv(0, 0) = b, H (0) = k > 0, Fuu(0, 0) = p, Fuv(0, 0) = q, Fvv(0, 0) = r, H (0) = l.
(3.3)
From (A2), it is known that (u, v) = (0, 0) is a trivial solution of (3.2) for any d, τ > 0. Let X = W 2,p(Ω) × W01,p(Ω) for p > n, and let Y = Lp(Ω). For the bifurcation of positive solutions of (3.2), fixing τ > 0, we define a nonlinear mapping W : R × X2 → Y 2 by
dΔu + F (u, v) W (d, u, v) = dΔv + 1 (H(u) − v) .
τ
Then, a solution (d, u, v) of (3.2) is equivalent to W (d, u, v) = (0, 0)T . Our main result on the local bifurcation of positive solutions of (3.2) is as follows:
(3.4)
Theorem 3.1. Suppose that τ > 0 is fixed, the conditions (A1) and (A2) hold, and also
(A3) a + bk > 0.
Define
d∗(τ ) = 1 (aτ − 1 + 2λ1τ
(aτ + 1)2 + 4bτ k),
(3.5)
where λ1 is the principal eigenvalue of −Δ in H01(Ω) with corresponding eigenfunction φ1(x) > 0. Then,
1. d = d∗ = d∗(τ ) is the unique bifurcation point of the system (3.2) where positive solutions of (3.2)
bifurcate from the line of trivial solutions Γ0 = {(d, 0, 0) : d > 0}.
43 Page 10 of 26
W. Zuo and J. Shi
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2. Near (d, u, v) = (d∗, 0, 0), there exists δ1 > 0 such that all positive solutions of (3.2) near the bifurcation point lie on a smooth curve Γ1 = {(d(s), ud(s, ·), vd(s, ·)) : s ∈ (0, δ1)} with d(s) = d∗ + d (0)s + s2z0(s), (ud(s, ·), vd(s, ·)) = s(1, M )φ1(·) + s2(z1(s, ·), z2(s, ·)), where
M= aτ + 1 +
2k ,
(aτ + 1)2 + 4bτ k
(3.6)
such that z0 : (0, δ1) → R and z1, z2 : (0, δ1) → X are smooth functions satisfying zi(0) = 0 for i = 0, 1, 2. Moreover,
[k(p + 2qM + rM 2) + bM l] φ31(x)dx
d (0) =
Ω
.
2λ1(k + M 2bτ ) φ21(x)dx
Ω
(3.7)
Proof. Let W be defined as in (3.4). Then, from (A1), W is twice differentiable in R × Xδ2, where Xδ is an open neighborhood of 0 in X. The Fr´echet derivative of W in variable (u, v) is
ξ1
dΔξ1 + Fu(u, v)ξ1 + Fv(u, v)ξ2
W(u,v)(d, u, v) ξ2 =
dΔξ2 + 1 (H (u)ξ1 − ξ2)
,
τ
and in particular when (u, v) = (0, 0),
(3.8)
W(u,v)(d, 0, 0) ξξ12 = ddΔΔξξ12 + A ξξ12 ,
(3.9)
where A is defined by
ab A = k −1 .
ττ The eigenvalues of A satisfy the characteristic equation
(3.10)
μ2 − a − 1 μ − a + bk = 0.
τ
τ
From (A3), we have a + bk > 0, then it is easy to see that A has a unique positive eigenvalue μ1 > 0
defined by
μ1 = 1 (aτ − 1 + (aτ + 1)2 + 4bτ k) 2τ
(3.11)
with a positive eigenvector (1, M ) where M is defined in (3.6). From the implicit function theorem, if d > 0
is a bifurcation point for positive solutions of (3.2) from the line of trivial solutions, then W(u,v)(d, 0, 0) is not invertible. That is, the null space N (W(u,v)(d, 0, 0)) = {0}. From Fourier theory, we must have d = μ1/λn, where λn is an eigenvalue of −Δ in H01(Ω). Since φ1 is the only eigenfunction which does not change sign in Ω, the only possible bifurcation point for positive solutions is d = d∗ = μ1/λ1 which is
given by (3.5). At (d∗, 0, 0), it is easy to compute the kernels of the linearized operator W(u,v)(d∗, 0, 0) and associated
adjoint operator W(∗u,v)(d∗, 0, 0), respectively:
N (W(u,v)(d∗, 0, 0)) = span{(1, M )φ1}, N (W(∗u,v)(d∗, 0, 0)) = span{(1, M bτ /k)φ1}.
And the range of the operator W(u,v)(d∗, 0, 0) is described by the following form:
⎧
⎫
⎨
⎬
R(W(u,v)(d∗, 0, 0)) = ⎩(g1, g2) ∈ Y 2 : (kg1(x) + M bτ g2(x)) φ1(x)dx = 0⎭ .
Ω