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On the stability of reaction–diffusion models with nonlocal delay effect and nonlinear boundary condition
Applied Mathematics Letters 103 (2020) 106197
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Applied Mathematics Letters www.elsevier.com/locate/aml
On the stability of reaction–diffusion models with nonlocal delay effect and nonlinear boundary condition✩ Shangjiang Guo a , Shangzhi Li a,b ,∗ a b
School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei, 430074, PR China College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China
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info
Article history: Received 18 October 2019 Received in revised form 18 December 2019 Accepted 18 December 2019 Available online 23 December 2019
abstract The local stability of the steady state for a general class of a reaction–diffusion model with nonlocal delay effect and nonlinear boundary condition is investigated under natural and minimal hypotheses. © 2019 Published by Elsevier Ltd.
Keywords: Reaction–diffusion Nonlocal delay effect Stability
1. Introduction In recent years, the existence, multiplicity, and uniqueness of positive solutions of nonlinear elliptic equations with nonlinear boundary conditions have been considered by many authors; See, for example, [1–3]. When time delay appears in the boundary forces in a nonlinear way, however, not much is known. Arag˜ao and Oliva [4] investigated a parabolic problem with nonlinear boundary conditions and delay in the boundary by constructing a reaction–diffusion problem with delay acting in the interior. Goddard II, Lee, and Shivaji [5] utilized the method of sub-supersolutions to investigate the existence of positive solution for a population dynamics model with strong Allee effect and nonlinear boundary conditions. In this paper, we are interested in the combined effect of the interior reaction delay and the boundary reaction delay on the dynamic behavior of the following parabolic problem with nonlinear boundary condition ⎧ ∂ ⎪ ⎨ u(x, t) =D∆u(x, t) + f (x, u(x, t), u ˆt (x)) , x ∈ Ω, ∂t (1) ⎪ ⎩ ∂ u(x, t) =g (x, u ˆt (x)) , x ∈ ∂Ω , ∂n ✩ Research supported by the NSFC (Grant No. 11671123). ∗ Corresponding Author. E-mail addresses: [email protected] (S. Guo), [email protected] (S. Li).
https://doi.org/10.1016/j.aml.2019.106197 0893-9659/© 2019 Published by Elsevier Ltd.
S. Guo and S. Li / Applied Mathematics Letters 103 (2020) 106197
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for t > 0, where τ ≥ 0, ∆ is the Laplace operator, x ∈ RN (N ≥ 1) is the spatial variable, t ≥ 0 is the time, u(x, t) ∈ Rn , D = diag(d1 , . . . , dn ) with positive constants di (i = 1, 2, . . . , n), Ω is a connected bounded open domain in RN with a smooth boundary ∂Ω , n is the unit outer normal to ∂Ω , f ∈ C 1,ϵ (Ω ×Rn ×Rn , Rn ) and g ∈ C 1,ϵ (∂Ω × Rn , Rn ) for some 0 < ϵ < 1, ⎧∫ τ∫ ⎪ ⎪ K0 (x, y, θ)u(y, t − θ)dydθ, x ∈ Ω, ⎨ 0 Ω u ˆt (x) = ∫ τ ⎪ ⎪ ⎩ K1 (θ)u(x, t − θ)dθ, x ∈ ∂Ω , 0
and K0 is a measurable function on Ω × Ω × [0, τ ) with values in Rn×n , while K1 is a measurable function on [0, τ ) with values in Rn×n . Throughout the paper we always assume that f (x, ·) ∈ C 2 (R2n , Rn ) and ∫τ g(y, ·) ∈ C 3 (Rn , Rn ) for all x ∈ Ω and y ∈ ∂Ω , and that 0 K1 (θ)dθ = 1 and there exist p > N and M0 > 0 such that [∫ [∫ ∫ ]1/p [∫ ] ] p−1
τ
p/(p−1)
|K0 (x, y, θ)| Ω
0
dydθ
Ω
(p−1)/p
τ
dx
p/(p−1)
|K1 (θ)|
+
dθ
< M0
0
for all x ∈ Ω . Hence, system (1) includes a reaction–diffusion with no-flux boundary condition (see, for example, [6]) and a boundary reaction with diffusion in interior (see, for example, [7]) as special cases. In biology, u(x, t) represents the population density of a species at time t and location x; The temporally ∫τ ∫ delayed and spatially nonlocal term given by the integral 0 Ω K0 (x, y, θ)u(y, t − θ)dydθ reflects the combined effect of the survivability and mobility of the immature individuals, and the kernel function K(x, y, θ) accounts for the probability that an individual born at location y can survive the immature period [0, τ ] and has moved to location x when becoming mature (τ time units after birth); The boundary condition means that the individuals are taken outside the habitat once they reach the boundary ∂Ω , at a rate which also depends on its population density in the time interval [t − τ, t], see [1] for a different nonlinearity on the boundary condition. Though nonlocal delay equations in mathematical biology have been studied extensively in [8,9] and references cited therein, to the best of our knowledge, the qualitative study of the nonlocal delay effect has not been found. The purpose of this note is to show how to obtain a linearization around a steady state of (1), and to give a proof of this result under natural and minimal hypotheses. As will be shown, the result is easy to understand and use. The result then provides the means to study the asymptotic stability of a steady state of a reaction–diffusion equation with nonlinear boundary conditions, and the question of Hopf bifurcation from the steady state. For convenience, we introduce the following notations. Denote by Lp (Ω ) (p ≥ N ) the Lebesgue space of integrable functions defined on Ω , and let W k,p (Ω ) (k ≥ 0) be the Sobolev space of the Lp -functions f (x) dn p 2,p (Ω ) defined on Ω whose derivatives dx n f (n = 1, . . . , k) also belong to L (Ω ). Denote the spaces X = W p (p−1)/p,p and Y = L (Ω ) × W (∂Ω ). Denote by Cτ = C([−τ, 0], X) the Banach space of continuous mappings from [−τ, 0] into X equipped with the supremum norm ∥ϕ∥Cτ = sup{∥ϕ(θ)∥X : θ ∈ [−τ, 0]} for ϕ ∈ Cτ . The steady-state solutions of (1) satisfy −∆u(x) = f (x, u(x), u ˆ(x)) for x ∈ Ω , and ∂u(x) ∂n = g(x, u(x)) for x ∈ ∂Ω , where ∫ ∫ τ
u ˆ(x) =
K0 (x, y, θ)u(y)dydθ, 0
x ∈ Ω.
Ω
Existence and nonexistence of steady-state solutions can be established by using a variational technique [10], the topological degree theory [11], Lyapunov–Schmidt reduction [12], and so on. In this paper, we assume system (1) has a steady state solution u∗ and obtain the following result on the local stability of system (1) at u = u∗ .
S. Guo and S. Li / Applied Mathematics Letters 103 (2020) 106197
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Theorem 1.1. Let A denote the infinitesimal generator of the semigroup generated by the linearized system of system (1) at a steady state solution u = u∗ , and σ(A) be the spectral set of A. (i) If sup{Reµ : µ ∈ σ(A)} = −α < 0, then the steady state solution u∗ of (1) is locally asymptotically stable. Namely, for each small ε > 0, there exist a constant M (ε) and a neighborhood V (ε) ⊂ Cτ of the steady state solution u∗ such that for ϕ ∈ V (ε), the solution u(x, t; ϕ) of (1) is defined for t ≥ 0 and ∥ut (ϕ) − u∗ ∥Cτ ≤ M (ε) exp {(ε − α) t} ∥ϕ∥Cτ . (ii) If sup{Reµ : µ ∈ σ(A)} = α > 0, then the steady state solution u∗ of (1) is unstable. Namely, there is a δ > 0, such that for each ε > 0, there is a ϕ ∈ Cτ , ∥ϕ − u∗ ∥Cτ ≤ ε and tϕ > 0 such that u(x, t; ϕ) exists for 0 ≤ t ≤ tϕ and ∥utϕ (ϕ) − u∗ ∥Cτ ≥ δ. In [2], Umezu investigated the existence, uniqueness, and stability of steady-state solution of the following initial boundary value problem: ⎧ ∂ ⎪ ⎨ u(x, t) =D∆u(x, t) + m(x)u(x, t) − au2 (x, t), x ∈ Ω, ∂t (2) ⎪ ⎩ ∂ u(x, t) = − u(x, t)g (u(x, t)) , x ∈ ∂Ω , ∂n where m(x) represents the local growth rate for some species, a is the effect of crowding for the species, the boundary condition means that the rate of the inflow of the population to the region Ω is governed by −u(x, t)g (u(x, t)). As stated in [2], the local stability of positive steady-state solution u∗ of (2) is determined by the largest eigenvalue µ1 of the following eigenvalue problem at a solution u = u∗ : µ1 v = ∂ v−[u∗ g ′ (u∗ ) + g(u∗ )] v on ∂Ω . By this means, Umezu investigated ∆v+[m(x) − 2au∗ ] v in Ω , and µ1 v = − ∂n many kinds of diffusive logistic equations with nonlinear boundary conditions (see, for example, [3]). As we shall see in Theorem 2.1, µ1 has the sign to that of the largest eigenvalue µ ˜1 of the eigenvalue problem: ∂ µ1 v = ∆v + [m(x) − 2au∗ ] v in Ω , and 0 = − ∂n v − [u∗ g ′ (u∗ ) + g(u∗ )] v on ∂Ω . Therefore, the conclusions obtained by Umezu [2,3] are obviously corollaries of our main result (Theorem 1.1). 2. Linearized system The linearization of system (1) at a steady state solution u = u∗ is as follows: ⎧ ∂ ⎪ ⎨ v(x, t) =∆v(x, t) + f0 (x)v(x, t) + f1 (x)ˆ ut (x), x ∈ Ω, ∂t ⎪ ⎩ ∂ v(x, t) =g0 (x)ˆ ut (x), x ∈ ∂Ω , ∂n ⏐ ⏐ ⏐ u∗ ) ⏐ ∂f (x,u∗ ,v) ⏐ ∂g(x,u) ⏐ where f0 (x) = ∂f (x,u,ˆ , f (x) = , g (x) = ⏐ ⏐ 1 0 ∂u ∂v ∂u ⏐ ∗ ∗ ∗ u=u (x)
u=ˆ u (x)
(3)
. The characteristic
u=u (x)
equation for the linear system (3) is obtained by considering solutions of the form v(x, t) = u(x) exp{µt} with u ∈ X. Let A be the infinitesimal generator of the semigroup generated by system (3). Namely, (Aψ)(θ) = def ˙ ψ(θ) for θ ∈ (−τ, 0) and (Aψ)(0) = ∆ψ(0) + f0 (·)ψ(0) + f1 (x)ψˆ for all ψ ∈ Cτ1 = {φ ∈ Cτ : φ˙ ∈ Cτ } ∂ satisfying ∂n ψ(0) = g0 (·)ψˆ on ∂Ω . Consider the following axillary system v(t) ˙ = F (vt , υ),
v0 = ϕ ∈ Cτ ,
where vt ∈ Cτ is defined by vt (θ) = v(t + θ) for θ ∈ [−τ, 0], and F : Cτ × (0, ∞) → Y is defined as ⎡ ⎤ ( ) ∆ϕ(0) + f x, ϕ(0) + u∗ , ϕˆ + u ˆ∗ − f (x, u∗ , u ˆ∗ ) ⎢ ⎥ F (ϕ, υ) = ⎣ ) 1 ⎦. 1 ∂ 1 ( ˆ ∗ ∗ − · ϕ(0) + g x, ϕ + u − g (x, u ) υ ∂n υ υ
(4)
4
S. Guo and S. Li / Applied Mathematics Letters 103 (2020) 106197
The linearized system of (4) at v = 0 is v(t) ˙ = Lυ vt , where ⎡ ⎤ ∆ϕ(0) + f0 (x)ϕ(0) + f1 (x)ϕˆ ⎦. Lυ ϕ = ⎣ 1 ∂ 1 − · ϕ(0) + g0 (x)ϕˆ υ ∂n υ For (µ, υ) ∈ C × R+ , define L(µ, υ): X → Y as [ ] ∫τ ∫ ∆u + f0 (·)u − µu + f1 (x) 0 Ω K0 (x, y, θ)u(y)e−µθ dydθ ∫τ L(µ, υ)u = . ∂u −µθ dθ − υµu ∂n − g0 (·)u 0 K1 (θ)e According to the bilinear form ⟨·, ·⟩: Y∗ × Y → R, we may define L∗ (µ, υ): Y∗ → X∗ by ⟨L∗ (µ, υ)u, v⟩ = ⟨u, L(µ, υ)v⟩ for all u ∈ Y∗ , v ∈ Y. In fact, we have [ ] ∫τ ∫ −µθ ∆u + f (·)u − µu + f (x)K (x, ·, θ)u(x)e dxdθ 0 1 0 0 Ω ∫τ L∗ (µ, υ)u = . ∂u − g (·)u K1 (θ)e−µθ dθ − υµu 0 ∂n 0 Thus, we conclude that µ ∈ σ(A) if and only if there exists u ∈ XC \ {0} such that L(µ, 0)u = 0, which is also equivalent to that there exists v ∈ XC \ {0} such that L∗ (µ, 0)v = 0. Let Aυ denote the infinitesimal generator of the semigroup {Tυ (t) : t ≥ 0} generated by system v(t) ˙ = Lυ vt . Similarly, we conclude that µ ∈ σ(Aυ ) if and only if there exists u ∈ XC \ {0} such that L(µ, υ)u = 0. Theorem 2.1. Assume that the linear operator L(µ0 , 0) has an eigenvalue 0 with an associated eigenvector q0 ∈ / RanL(µ0 , 0), then there exist a positive constant δ, a continuously differentiable mapping υ ↦→ µ(υ) from (−δ, δ) into C, and a continuously differentiable mapping υ ↦→ q(υ) from (−δ, δ) into X, such that µ(0) = µ0 , q(0) = q0 , and L(µ(υ), υ)q(υ) = 0 for υ ∈ (−δ, δ). (5) Proof . The assumption means that we have the following direct sum decomposition: Y = KerL∗ (µ0 , 0) ⊕ RanL(µ0 , 0), which induce a decomposition X = KerL(µ0 , 0) ⊕ [X ∩ RanL∗ (µ0 , 0)]. Define a mapping G: R × [X ∩ RanL∗ (µ0 , 0)] × C → Y by G(υ, ϕ, µ) = L(µ, υ)(q0 + ϕ). Then G(0, 0, µ0 ) = 0, Dϕ G(0, 0, µ0 ) = L(µ0 , 0), and [ ] ∫τ ∫ −q0 − f1 (·) 0 Ω K0 (x, y, θ)q0 (y)θe−µθ dydθ ∫τ Dµ G(0, 0, µ0 ) = . g0 (·)q0 0 θK1 (θ)e−µθ dθ Thus, the assumption implies that D(ϕ,µ) G(0, 0, µ0 ) : [X ∩ RanL∗ (µ0 , 0)] × C → Y is an isomorphism. The implicit function theorem then gives continuously differentiable functions (ϕ, µ): (−δ, δ) → [X ∩ RanL∗ (µ0 , 0)] × C such that ϕ(0) = 0, µ(0) = µ0 , and G(υ, ϕ(υ), µ(υ)) = 0 for all υ ∈ (−δ, δ). Therefore, we have (5) with q(υ) = q0 + ϕ(υ). Thus, the proof is complete. □ In particular, if L(µ0 , 0) has a simple eigenvalue 0 of L(µ0 , 0) then the eigenvector q0∗ of L∗ (µ0 , 0) with eigenvalue 0 can be chosen so that ⟨q0∗ , q0 ⟩ = 1, ⟨q0∗ , q¯0 ⟩ = 1, and RanL(µ0 , 0) = {z ∈ Y : ⟨q0∗ , z⟩ = 0}. Differentiating both sides of (5) yields µ′ (0)Dµ L(µ0 , 0)q0 + Dυ L(µ0 , 0)q0 + L(µ0 , 0)q ′ (0) = 0, and hence 0 = µ′ (0)⟨q0∗ , Dµ L(µ0 , 0)q0 ⟩ + ⟨q0∗ , Dυ L(µ0 , 0)q0 ⟩, which implies that µ′ (0) = −⟨q0∗ , Dυ L(µ0 , 0)q0 ⟩/⟨q0∗ , Dµ L(µ0 , 0)q0 ⟩. Remark 2.1. Theorem 2.1 implies that if A has an m-multiple eigenvalue µ then for every sufficiently small positive constant υ, Aυ has an m-multiple eigenvalue µ(υ), which is continuously differentiable with respect to υ. We know that there exist only a finite number of eigenvalues of A (as well as Aυ ) in any vertical strip in the complex plane. This implies that A (or Aυ ) has an eigenvalue with the largest real parts. Thus, if all
S. Guo and S. Li / Applied Mathematics Letters 103 (2020) 106197
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the eigenvalues of A have negative real parts (i.e., conditions of Theorem 1.1(i) are satisfied) then all the eigenvalues of Aυ with sufficiently small positive constant υ have negative real parts. If A has one eigenvalue with a negative real part (i.e., conditions of Theorem 1.1(ii) are satisfied) then Aυ with sufficiently small positive constant υ has one eigenvalue with a negative real part. 3. Local stability of the auxiliary system (4) Lemma 3.1. Let G(ϕ, υ) = F (ϕ, υ) − Lυ ϕ. Then there exists a constant N (υ, τ ) > 0 such that for each ν ∈ (0, 1), there is an h(ν) > 0 such that for ϕ ∈ Bh(ν) ≜ {φ ∈ Cτ : ∥φ∥ ≤ h(ν)}, we have ∥G(ϕ, υ)∥Y ≤ νN (υ, τ )∥ϕ∥Cτ . Proof . For all x ∈ Ω , we have [∫ ⏐ ]1/p [∫ [∫ ⏐ ⏐ ˆ ⏐p ≤ ⏐ϕ(x)⏐ dx Ω
Ω
and [∫ ∂Ω
0
τ
∫
q
]1/p [∫
]p/q
|K0 (x, y, θ)| dydθ
0
q
]1/q [∫
∫
|K1 (θ)| dθ
,
Ω
τ
p
]1/p
|ϕ(x, −θ)| dθdσ ∂Ω
]1/p
p
|ϕ(y, −θ)| dydθ 0
τ
∫
dx
Ω
]1/p [∫ ⏐ ⏐ ⏐ ˆ ⏐p ≤ ⏐ϕ(x)⏐ dσ
τ
,
0
ˆ Y ≤ M0 τ ∥ϕ∥C for all ϕ ∈ Cτ . By continuous where p > N and q = p/(p − 1). This implies that ∥ϕ∥ τ differentiability of f and g for each ν ∈ (0, 1], we can choose an h(ν) > 0, such that for ϕ ∈ Bh(ν) , ⏐ ⏐ ∂f (x, u∗ + u, u ˆ∗ ) ⏐⏐ ∂f (x, ϕ(0) + u∗ , u ˆ∗ + v) ⏐⏐ −ν ≤ − f (x) ≤ ν, −ν ≤ 0 ⏐ ⏐ ˆ − f1 (x) ≤ ν ∂u ∂v u=ϕ(0) v=ϕ for x ∈ Ω , and −ν ≤
⏐ ∂g(x, u∗ + v) ⏐⏐ ⏐ ˆ − g0 (x) ≤ ν ∂v v=ϕ
for x ∈ ∂Ω .
Thus, for ϕ ∈ Bh(ν) we have ⏐ ( ⏐ ) ⏐ ⏐ ˆ∗ − f (x, u∗ , u ˆ∗ ) − f0 (x)ϕ(0) − f1 (x)ϕˆ⏐ ⏐f x, ϕ(0) + u∗ , ϕˆ + u ⏐ ⏐ ⏐ ⏐ ∂f (x, u∗ + u, u ⏐ ˆ∗ ) ⏐⏐ ⏐ ⏐ ≤⏐ − f0 (x)⏐ · |ϕ(0)| ⏐ ⏐ ⏐ ∂u u=θϕ(0) ⏐ ⏐ ⏐ ⏐ ∂f (x, ϕ(0) + u∗ , u ⏐ ˆ∗ + v) ⏐⏐ ⏐ ⏐ ˆ ˆ + ⏐ − f1 (x)⏐ · |ϕ| ≤ ν|ϕ(0)| + ν|ϕ| ⏐ ⏐ ⏐ ∂v ˆ v=ξ ϕ in Ω , and
⏐ ⏐ ⏐ ( ⏐ ⏐ ∂f (x, u∗ + v) ⏐⏐ ⏐ ) ⏐ ⏐ ⏐ ˆ ⏐ ˆ ⏐ − g (x) ⏐g x, ϕˆ + u∗ − g (x, u∗ ) − g0 (x)ϕˆ⏐ ≤ ⏐ ⏐ · |ϕ| ≤ ν|ϕ| 0 ⏐ ⏐ ⏐ ∂v ˆ v=ζ ϕ
on ∂Ω , where θ, ξ, and ζ lie in (0, 1). Therefore, [∫ ⏐ ( ⏐p ]1/p ) ⏐ ⏐ ∥G(ϕ, υ)∥Y = ˆ∗ − f (x, u∗ , u ˆ∗ ) − f0 (x)ϕ(0) − f1 (x)ϕˆ⏐ dx ⏐f x, ϕ(0) + u∗ , ϕˆ + u Ω
[∫ ⏐ ( ⏐p ]1/p ) 1 ⏐ ⏐ ⏐g x, ϕˆ + u∗ − g (x, u∗ ) − g0 (x)ϕˆ⏐ dσ υ ∂Ω [∫ ( )p ]1/p ν [∫ ⏐ ⏐p ]1/p ⏐ ˆ⏐ ˆ dx ≤ν |ϕ(0)| + |ϕ| + , ⏐ϕ⏐ dσ υ ∂Ω Ω +
Namely, we have ∥G(ϕ, υ)∥Y ≤ νN (υ, τ )∥ϕ∥Cτ with N (υ, τ ) = (1 + υ)(1 + M0 τ )/υ. The proof is completed. □
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S. Guo and S. Li / Applied Mathematics Letters 103 (2020) 106197
We may enlarge the phase space Cτ such that (4) can be rewritten as an abstract ordinary differential equation (ODE) in the Banach space BCτ . Firstly, in BCτ , we consider an extension of the infinitesimal generator Aυ , still denoted by Aυ , Aυ : BCτ ⊃ Cτ1 ∋ φ ↦→ φ˙ + X0 [Lυ φ − φ] ˙ ∈ BCτ , where Dom(Aυ ) = Cτ1 . d u = Aυ u + X0 G(u, υ). Thus, the abstract ODE in BCτ associated with (4) can be rewritten in the form dt ∫t By the formula of variation of constants for t ≥ τ we have ut (ϕ) = Tυ (t)ϕ + 0 Tυ (t − s)X0 G(us (ϕ), υ)ds and hence ∫ t (6) ∥ut (ϕ)∥Cτ ≤ ∥Tυ (t)∥ · ∥ϕ∥Cτ + νN (υ, τ ) ∥Tυ (t − s)∥ · ∥us (ϕ)∥Cτ ds. 0
Thus, we can obtain the following local stability result of the trivial solution of (4). Theorem 3.1. If sup{Reµ : µ ∈ σ(Aυ )} = −α < 0, then the zero solution of (4) is locally asymptotically stable; If sup{Reµ : µ ∈ σ(Aυ )} = α > 0, then the zero solution of (4) is unstable. Proof . We start with the case where sup{Reµ : µ ∈ σ(Aυ )} = −α < 0. Then for each given ε ∈ (0, α], ) } {( there exists a constant M1 (ε) > 1 such that ∥Tυ (t)ϕ∥Cτ ≤ M1 (ε) exp 2ε − α t ∥ϕ∥Cτ for all t ≥ 0 and ϕ ∈ Cτ . It follows from (6) that ∥ut (ϕ)∥Cτ ≤M1 (ε) exp
∫ t ) } ) } {( ε − α t ∥ϕ∥Cτ + νN (υ, τ )M1 (ε) − α (t − s) ∥us (ϕ)∥Cτ ds. exp 2 2 0
{( ε
By applying the Gronwall’s inequality we have ∥ut (ϕ)∥Cτ ≤ M1 (ε) exp {(ε − α) t} ∥ϕ∥Cτ for all t ≥ τ . In what follows, we consider the case where sup{Reµ : µ ∈ σ(Aυ )} = α > 0. Using the same estimate as the above discussion and a similar argument to the usual functional differential equations (for instance, see Theorem 1.1, Chapter 10 in [13]), we are able to show that Eq. (4) has a nonzero solution u(x, t) which is defined for all t ≤ 0 and u(·, t) → 0 as t → −∞. Thus the zero solution is unstable. The proof is completed. □ Proof of Theorem 1.1. If sup{Reµ : µ ∈ σ(A)} = −α < 0, then there exists a sufficiently small υ > 0 such that sup{Reµ : µ ∈ σ(Aυ )} = − α2 < 0. This implies that for each small ε > 0, there exist a constant M (ε) and a neighborhood V (ε) ⊂ Cτ of the origin such that for ϕ ∈ V (ε), the solution v(t; ϕ, υ) of (4) is defined {( ) } for t ≥ 0 and ∥vt (ϕ, υ)∥Cτ ≤ M (ε) exp ε − α2 t ∥ϕ∥Cτ . Note that v(t; ϕ, υ) is continuous with respect to ) } {( υ, then we see that ∥vt (ϕ, 0)∥Cτ ≤ M (ε) exp ε − α2 t ∥ϕ∥Cτ , where v(t; ϕ, 0) is a solution of the following system with initial value ϕ: ⎧ ˙ =∆u(t) + f (x, u(t) + u∗ , u ˆt + u ˆ∗ ) − f (x, u∗ , u ˆ∗ ) , ⎨ u(t) ⎩
0=−
∂ u(t) + g (x, u ˆt + u∗ ) − g (x, u∗ ) . ∂n
which implies that the steady state solution u∗ of (1) is locally asymptotically stable. Similarly, we can show that the steady state solution u∗ of (1) is unstable if sup{Reµ : µ ∈ σ(A)} = α > 0. Thus, the proof of Theorem 1.1 is completed. □ CRediT authorship contribution statement Shangjiang Guo: Supervision, Conceptualization, Writing - review & editing, Funding acquisition. Shangzhi Li: Formal analysis, Writing - original draft.
S. Guo and S. Li / Applied Mathematics Letters 103 (2020) 106197
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Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
On the stability of reaction–diffusion models with nonlocal delay effect and nonlinear boundary condition✩ Shangjiang Guo a , Shangzhi Li a,b ,∗ a b
School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei, 430074, PR China College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China
article
info
Article history: Received 18 October 2019 Received in revised form 18 December 2019 Accepted 18 December 2019 Available online 23 December 2019
abstract The local stability of the steady state for a general class of a reaction–diffusion model with nonlocal delay effect and nonlinear boundary condition is investigated under natural and minimal hypotheses. © 2019 Published by Elsevier Ltd.
Keywords: Reaction–diffusion Nonlocal delay effect Stability
1. Introduction In recent years, the existence, multiplicity, and uniqueness of positive solutions of nonlinear elliptic equations with nonlinear boundary conditions have been considered by many authors; See, for example, [1–3]. When time delay appears in the boundary forces in a nonlinear way, however, not much is known. Arag˜ao and Oliva [4] investigated a parabolic problem with nonlinear boundary conditions and delay in the boundary by constructing a reaction–diffusion problem with delay acting in the interior. Goddard II, Lee, and Shivaji [5] utilized the method of sub-supersolutions to investigate the existence of positive solution for a population dynamics model with strong Allee effect and nonlinear boundary conditions. In this paper, we are interested in the combined effect of the interior reaction delay and the boundary reaction delay on the dynamic behavior of the following parabolic problem with nonlinear boundary condition ⎧ ∂ ⎪ ⎨ u(x, t) =D∆u(x, t) + f (x, u(x, t), u ˆt (x)) , x ∈ Ω, ∂t (1) ⎪ ⎩ ∂ u(x, t) =g (x, u ˆt (x)) , x ∈ ∂Ω , ∂n ✩ Research supported by the NSFC (Grant No. 11671123). ∗ Corresponding Author. E-mail addresses: [email protected] (S. Guo), [email protected] (S. Li).
https://doi.org/10.1016/j.aml.2019.106197 0893-9659/© 2019 Published by Elsevier Ltd.
S. Guo and S. Li / Applied Mathematics Letters 103 (2020) 106197
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for t > 0, where τ ≥ 0, ∆ is the Laplace operator, x ∈ RN (N ≥ 1) is the spatial variable, t ≥ 0 is the time, u(x, t) ∈ Rn , D = diag(d1 , . . . , dn ) with positive constants di (i = 1, 2, . . . , n), Ω is a connected bounded open domain in RN with a smooth boundary ∂Ω , n is the unit outer normal to ∂Ω , f ∈ C 1,ϵ (Ω ×Rn ×Rn , Rn ) and g ∈ C 1,ϵ (∂Ω × Rn , Rn ) for some 0 < ϵ < 1, ⎧∫ τ∫ ⎪ ⎪ K0 (x, y, θ)u(y, t − θ)dydθ, x ∈ Ω, ⎨ 0 Ω u ˆt (x) = ∫ τ ⎪ ⎪ ⎩ K1 (θ)u(x, t − θ)dθ, x ∈ ∂Ω , 0
and K0 is a measurable function on Ω × Ω × [0, τ ) with values in Rn×n , while K1 is a measurable function on [0, τ ) with values in Rn×n . Throughout the paper we always assume that f (x, ·) ∈ C 2 (R2n , Rn ) and ∫τ g(y, ·) ∈ C 3 (Rn , Rn ) for all x ∈ Ω and y ∈ ∂Ω , and that 0 K1 (θ)dθ = 1 and there exist p > N and M0 > 0 such that [∫ [∫ ∫ ]1/p [∫ ] ] p−1
τ
p/(p−1)
|K0 (x, y, θ)| Ω
0
dydθ
Ω
(p−1)/p
τ
dx
p/(p−1)
|K1 (θ)|
+
dθ
< M0
0
for all x ∈ Ω . Hence, system (1) includes a reaction–diffusion with no-flux boundary condition (see, for example, [6]) and a boundary reaction with diffusion in interior (see, for example, [7]) as special cases. In biology, u(x, t) represents the population density of a species at time t and location x; The temporally ∫τ ∫ delayed and spatially nonlocal term given by the integral 0 Ω K0 (x, y, θ)u(y, t − θ)dydθ reflects the combined effect of the survivability and mobility of the immature individuals, and the kernel function K(x, y, θ) accounts for the probability that an individual born at location y can survive the immature period [0, τ ] and has moved to location x when becoming mature (τ time units after birth); The boundary condition means that the individuals are taken outside the habitat once they reach the boundary ∂Ω , at a rate which also depends on its population density in the time interval [t − τ, t], see [1] for a different nonlinearity on the boundary condition. Though nonlocal delay equations in mathematical biology have been studied extensively in [8,9] and references cited therein, to the best of our knowledge, the qualitative study of the nonlocal delay effect has not been found. The purpose of this note is to show how to obtain a linearization around a steady state of (1), and to give a proof of this result under natural and minimal hypotheses. As will be shown, the result is easy to understand and use. The result then provides the means to study the asymptotic stability of a steady state of a reaction–diffusion equation with nonlinear boundary conditions, and the question of Hopf bifurcation from the steady state. For convenience, we introduce the following notations. Denote by Lp (Ω ) (p ≥ N ) the Lebesgue space of integrable functions defined on Ω , and let W k,p (Ω ) (k ≥ 0) be the Sobolev space of the Lp -functions f (x) dn p 2,p (Ω ) defined on Ω whose derivatives dx n f (n = 1, . . . , k) also belong to L (Ω ). Denote the spaces X = W p (p−1)/p,p and Y = L (Ω ) × W (∂Ω ). Denote by Cτ = C([−τ, 0], X) the Banach space of continuous mappings from [−τ, 0] into X equipped with the supremum norm ∥ϕ∥Cτ = sup{∥ϕ(θ)∥X : θ ∈ [−τ, 0]} for ϕ ∈ Cτ . The steady-state solutions of (1) satisfy −∆u(x) = f (x, u(x), u ˆ(x)) for x ∈ Ω , and ∂u(x) ∂n = g(x, u(x)) for x ∈ ∂Ω , where ∫ ∫ τ
u ˆ(x) =
K0 (x, y, θ)u(y)dydθ, 0
x ∈ Ω.
Ω
Existence and nonexistence of steady-state solutions can be established by using a variational technique [10], the topological degree theory [11], Lyapunov–Schmidt reduction [12], and so on. In this paper, we assume system (1) has a steady state solution u∗ and obtain the following result on the local stability of system (1) at u = u∗ .
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Theorem 1.1. Let A denote the infinitesimal generator of the semigroup generated by the linearized system of system (1) at a steady state solution u = u∗ , and σ(A) be the spectral set of A. (i) If sup{Reµ : µ ∈ σ(A)} = −α < 0, then the steady state solution u∗ of (1) is locally asymptotically stable. Namely, for each small ε > 0, there exist a constant M (ε) and a neighborhood V (ε) ⊂ Cτ of the steady state solution u∗ such that for ϕ ∈ V (ε), the solution u(x, t; ϕ) of (1) is defined for t ≥ 0 and ∥ut (ϕ) − u∗ ∥Cτ ≤ M (ε) exp {(ε − α) t} ∥ϕ∥Cτ . (ii) If sup{Reµ : µ ∈ σ(A)} = α > 0, then the steady state solution u∗ of (1) is unstable. Namely, there is a δ > 0, such that for each ε > 0, there is a ϕ ∈ Cτ , ∥ϕ − u∗ ∥Cτ ≤ ε and tϕ > 0 such that u(x, t; ϕ) exists for 0 ≤ t ≤ tϕ and ∥utϕ (ϕ) − u∗ ∥Cτ ≥ δ. In [2], Umezu investigated the existence, uniqueness, and stability of steady-state solution of the following initial boundary value problem: ⎧ ∂ ⎪ ⎨ u(x, t) =D∆u(x, t) + m(x)u(x, t) − au2 (x, t), x ∈ Ω, ∂t (2) ⎪ ⎩ ∂ u(x, t) = − u(x, t)g (u(x, t)) , x ∈ ∂Ω , ∂n where m(x) represents the local growth rate for some species, a is the effect of crowding for the species, the boundary condition means that the rate of the inflow of the population to the region Ω is governed by −u(x, t)g (u(x, t)). As stated in [2], the local stability of positive steady-state solution u∗ of (2) is determined by the largest eigenvalue µ1 of the following eigenvalue problem at a solution u = u∗ : µ1 v = ∂ v−[u∗ g ′ (u∗ ) + g(u∗ )] v on ∂Ω . By this means, Umezu investigated ∆v+[m(x) − 2au∗ ] v in Ω , and µ1 v = − ∂n many kinds of diffusive logistic equations with nonlinear boundary conditions (see, for example, [3]). As we shall see in Theorem 2.1, µ1 has the sign to that of the largest eigenvalue µ ˜1 of the eigenvalue problem: ∂ µ1 v = ∆v + [m(x) − 2au∗ ] v in Ω , and 0 = − ∂n v − [u∗ g ′ (u∗ ) + g(u∗ )] v on ∂Ω . Therefore, the conclusions obtained by Umezu [2,3] are obviously corollaries of our main result (Theorem 1.1). 2. Linearized system The linearization of system (1) at a steady state solution u = u∗ is as follows: ⎧ ∂ ⎪ ⎨ v(x, t) =∆v(x, t) + f0 (x)v(x, t) + f1 (x)ˆ ut (x), x ∈ Ω, ∂t ⎪ ⎩ ∂ v(x, t) =g0 (x)ˆ ut (x), x ∈ ∂Ω , ∂n ⏐ ⏐ ⏐ u∗ ) ⏐ ∂f (x,u∗ ,v) ⏐ ∂g(x,u) ⏐ where f0 (x) = ∂f (x,u,ˆ , f (x) = , g (x) = ⏐ ⏐ 1 0 ∂u ∂v ∂u ⏐ ∗ ∗ ∗ u=u (x)
u=ˆ u (x)
(3)
. The characteristic
u=u (x)
equation for the linear system (3) is obtained by considering solutions of the form v(x, t) = u(x) exp{µt} with u ∈ X. Let A be the infinitesimal generator of the semigroup generated by system (3). Namely, (Aψ)(θ) = def ˙ ψ(θ) for θ ∈ (−τ, 0) and (Aψ)(0) = ∆ψ(0) + f0 (·)ψ(0) + f1 (x)ψˆ for all ψ ∈ Cτ1 = {φ ∈ Cτ : φ˙ ∈ Cτ } ∂ satisfying ∂n ψ(0) = g0 (·)ψˆ on ∂Ω . Consider the following axillary system v(t) ˙ = F (vt , υ),
v0 = ϕ ∈ Cτ ,
where vt ∈ Cτ is defined by vt (θ) = v(t + θ) for θ ∈ [−τ, 0], and F : Cτ × (0, ∞) → Y is defined as ⎡ ⎤ ( ) ∆ϕ(0) + f x, ϕ(0) + u∗ , ϕˆ + u ˆ∗ − f (x, u∗ , u ˆ∗ ) ⎢ ⎥ F (ϕ, υ) = ⎣ ) 1 ⎦. 1 ∂ 1 ( ˆ ∗ ∗ − · ϕ(0) + g x, ϕ + u − g (x, u ) υ ∂n υ υ
(4)
4
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The linearized system of (4) at v = 0 is v(t) ˙ = Lυ vt , where ⎡ ⎤ ∆ϕ(0) + f0 (x)ϕ(0) + f1 (x)ϕˆ ⎦. Lυ ϕ = ⎣ 1 ∂ 1 − · ϕ(0) + g0 (x)ϕˆ υ ∂n υ For (µ, υ) ∈ C × R+ , define L(µ, υ): X → Y as [ ] ∫τ ∫ ∆u + f0 (·)u − µu + f1 (x) 0 Ω K0 (x, y, θ)u(y)e−µθ dydθ ∫τ L(µ, υ)u = . ∂u −µθ dθ − υµu ∂n − g0 (·)u 0 K1 (θ)e According to the bilinear form ⟨·, ·⟩: Y∗ × Y → R, we may define L∗ (µ, υ): Y∗ → X∗ by ⟨L∗ (µ, υ)u, v⟩ = ⟨u, L(µ, υ)v⟩ for all u ∈ Y∗ , v ∈ Y. In fact, we have [ ] ∫τ ∫ −µθ ∆u + f (·)u − µu + f (x)K (x, ·, θ)u(x)e dxdθ 0 1 0 0 Ω ∫τ L∗ (µ, υ)u = . ∂u − g (·)u K1 (θ)e−µθ dθ − υµu 0 ∂n 0 Thus, we conclude that µ ∈ σ(A) if and only if there exists u ∈ XC \ {0} such that L(µ, 0)u = 0, which is also equivalent to that there exists v ∈ XC \ {0} such that L∗ (µ, 0)v = 0. Let Aυ denote the infinitesimal generator of the semigroup {Tυ (t) : t ≥ 0} generated by system v(t) ˙ = Lυ vt . Similarly, we conclude that µ ∈ σ(Aυ ) if and only if there exists u ∈ XC \ {0} such that L(µ, υ)u = 0. Theorem 2.1. Assume that the linear operator L(µ0 , 0) has an eigenvalue 0 with an associated eigenvector q0 ∈ / RanL(µ0 , 0), then there exist a positive constant δ, a continuously differentiable mapping υ ↦→ µ(υ) from (−δ, δ) into C, and a continuously differentiable mapping υ ↦→ q(υ) from (−δ, δ) into X, such that µ(0) = µ0 , q(0) = q0 , and L(µ(υ), υ)q(υ) = 0 for υ ∈ (−δ, δ). (5) Proof . The assumption means that we have the following direct sum decomposition: Y = KerL∗ (µ0 , 0) ⊕ RanL(µ0 , 0), which induce a decomposition X = KerL(µ0 , 0) ⊕ [X ∩ RanL∗ (µ0 , 0)]. Define a mapping G: R × [X ∩ RanL∗ (µ0 , 0)] × C → Y by G(υ, ϕ, µ) = L(µ, υ)(q0 + ϕ). Then G(0, 0, µ0 ) = 0, Dϕ G(0, 0, µ0 ) = L(µ0 , 0), and [ ] ∫τ ∫ −q0 − f1 (·) 0 Ω K0 (x, y, θ)q0 (y)θe−µθ dydθ ∫τ Dµ G(0, 0, µ0 ) = . g0 (·)q0 0 θK1 (θ)e−µθ dθ Thus, the assumption implies that D(ϕ,µ) G(0, 0, µ0 ) : [X ∩ RanL∗ (µ0 , 0)] × C → Y is an isomorphism. The implicit function theorem then gives continuously differentiable functions (ϕ, µ): (−δ, δ) → [X ∩ RanL∗ (µ0 , 0)] × C such that ϕ(0) = 0, µ(0) = µ0 , and G(υ, ϕ(υ), µ(υ)) = 0 for all υ ∈ (−δ, δ). Therefore, we have (5) with q(υ) = q0 + ϕ(υ). Thus, the proof is complete. □ In particular, if L(µ0 , 0) has a simple eigenvalue 0 of L(µ0 , 0) then the eigenvector q0∗ of L∗ (µ0 , 0) with eigenvalue 0 can be chosen so that ⟨q0∗ , q0 ⟩ = 1, ⟨q0∗ , q¯0 ⟩ = 1, and RanL(µ0 , 0) = {z ∈ Y : ⟨q0∗ , z⟩ = 0}. Differentiating both sides of (5) yields µ′ (0)Dµ L(µ0 , 0)q0 + Dυ L(µ0 , 0)q0 + L(µ0 , 0)q ′ (0) = 0, and hence 0 = µ′ (0)⟨q0∗ , Dµ L(µ0 , 0)q0 ⟩ + ⟨q0∗ , Dυ L(µ0 , 0)q0 ⟩, which implies that µ′ (0) = −⟨q0∗ , Dυ L(µ0 , 0)q0 ⟩/⟨q0∗ , Dµ L(µ0 , 0)q0 ⟩. Remark 2.1. Theorem 2.1 implies that if A has an m-multiple eigenvalue µ then for every sufficiently small positive constant υ, Aυ has an m-multiple eigenvalue µ(υ), which is continuously differentiable with respect to υ. We know that there exist only a finite number of eigenvalues of A (as well as Aυ ) in any vertical strip in the complex plane. This implies that A (or Aυ ) has an eigenvalue with the largest real parts. Thus, if all
S. Guo and S. Li / Applied Mathematics Letters 103 (2020) 106197
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the eigenvalues of A have negative real parts (i.e., conditions of Theorem 1.1(i) are satisfied) then all the eigenvalues of Aυ with sufficiently small positive constant υ have negative real parts. If A has one eigenvalue with a negative real part (i.e., conditions of Theorem 1.1(ii) are satisfied) then Aυ with sufficiently small positive constant υ has one eigenvalue with a negative real part. 3. Local stability of the auxiliary system (4) Lemma 3.1. Let G(ϕ, υ) = F (ϕ, υ) − Lυ ϕ. Then there exists a constant N (υ, τ ) > 0 such that for each ν ∈ (0, 1), there is an h(ν) > 0 such that for ϕ ∈ Bh(ν) ≜ {φ ∈ Cτ : ∥φ∥ ≤ h(ν)}, we have ∥G(ϕ, υ)∥Y ≤ νN (υ, τ )∥ϕ∥Cτ . Proof . For all x ∈ Ω , we have [∫ ⏐ ]1/p [∫ [∫ ⏐ ⏐ ˆ ⏐p ≤ ⏐ϕ(x)⏐ dx Ω
Ω
and [∫ ∂Ω
0
τ
∫
q
]1/p [∫
]p/q
|K0 (x, y, θ)| dydθ
0
q
]1/q [∫
∫
|K1 (θ)| dθ
,
Ω
τ
p
]1/p
|ϕ(x, −θ)| dθdσ ∂Ω
]1/p
p
|ϕ(y, −θ)| dydθ 0
τ
∫
dx
Ω
]1/p [∫ ⏐ ⏐ ⏐ ˆ ⏐p ≤ ⏐ϕ(x)⏐ dσ
τ
,
0
ˆ Y ≤ M0 τ ∥ϕ∥C for all ϕ ∈ Cτ . By continuous where p > N and q = p/(p − 1). This implies that ∥ϕ∥ τ differentiability of f and g for each ν ∈ (0, 1], we can choose an h(ν) > 0, such that for ϕ ∈ Bh(ν) , ⏐ ⏐ ∂f (x, u∗ + u, u ˆ∗ ) ⏐⏐ ∂f (x, ϕ(0) + u∗ , u ˆ∗ + v) ⏐⏐ −ν ≤ − f (x) ≤ ν, −ν ≤ 0 ⏐ ⏐ ˆ − f1 (x) ≤ ν ∂u ∂v u=ϕ(0) v=ϕ for x ∈ Ω , and −ν ≤
⏐ ∂g(x, u∗ + v) ⏐⏐ ⏐ ˆ − g0 (x) ≤ ν ∂v v=ϕ
for x ∈ ∂Ω .
Thus, for ϕ ∈ Bh(ν) we have ⏐ ( ⏐ ) ⏐ ⏐ ˆ∗ − f (x, u∗ , u ˆ∗ ) − f0 (x)ϕ(0) − f1 (x)ϕˆ⏐ ⏐f x, ϕ(0) + u∗ , ϕˆ + u ⏐ ⏐ ⏐ ⏐ ∂f (x, u∗ + u, u ⏐ ˆ∗ ) ⏐⏐ ⏐ ⏐ ≤⏐ − f0 (x)⏐ · |ϕ(0)| ⏐ ⏐ ⏐ ∂u u=θϕ(0) ⏐ ⏐ ⏐ ⏐ ∂f (x, ϕ(0) + u∗ , u ⏐ ˆ∗ + v) ⏐⏐ ⏐ ⏐ ˆ ˆ + ⏐ − f1 (x)⏐ · |ϕ| ≤ ν|ϕ(0)| + ν|ϕ| ⏐ ⏐ ⏐ ∂v ˆ v=ξ ϕ in Ω , and
⏐ ⏐ ⏐ ( ⏐ ⏐ ∂f (x, u∗ + v) ⏐⏐ ⏐ ) ⏐ ⏐ ⏐ ˆ ⏐ ˆ ⏐ − g (x) ⏐g x, ϕˆ + u∗ − g (x, u∗ ) − g0 (x)ϕˆ⏐ ≤ ⏐ ⏐ · |ϕ| ≤ ν|ϕ| 0 ⏐ ⏐ ⏐ ∂v ˆ v=ζ ϕ
on ∂Ω , where θ, ξ, and ζ lie in (0, 1). Therefore, [∫ ⏐ ( ⏐p ]1/p ) ⏐ ⏐ ∥G(ϕ, υ)∥Y = ˆ∗ − f (x, u∗ , u ˆ∗ ) − f0 (x)ϕ(0) − f1 (x)ϕˆ⏐ dx ⏐f x, ϕ(0) + u∗ , ϕˆ + u Ω
[∫ ⏐ ( ⏐p ]1/p ) 1 ⏐ ⏐ ⏐g x, ϕˆ + u∗ − g (x, u∗ ) − g0 (x)ϕˆ⏐ dσ υ ∂Ω [∫ ( )p ]1/p ν [∫ ⏐ ⏐p ]1/p ⏐ ˆ⏐ ˆ dx ≤ν |ϕ(0)| + |ϕ| + , ⏐ϕ⏐ dσ υ ∂Ω Ω +
Namely, we have ∥G(ϕ, υ)∥Y ≤ νN (υ, τ )∥ϕ∥Cτ with N (υ, τ ) = (1 + υ)(1 + M0 τ )/υ. The proof is completed. □
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We may enlarge the phase space Cτ such that (4) can be rewritten as an abstract ordinary differential equation (ODE) in the Banach space BCτ . Firstly, in BCτ , we consider an extension of the infinitesimal generator Aυ , still denoted by Aυ , Aυ : BCτ ⊃ Cτ1 ∋ φ ↦→ φ˙ + X0 [Lυ φ − φ] ˙ ∈ BCτ , where Dom(Aυ ) = Cτ1 . d u = Aυ u + X0 G(u, υ). Thus, the abstract ODE in BCτ associated with (4) can be rewritten in the form dt ∫t By the formula of variation of constants for t ≥ τ we have ut (ϕ) = Tυ (t)ϕ + 0 Tυ (t − s)X0 G(us (ϕ), υ)ds and hence ∫ t (6) ∥ut (ϕ)∥Cτ ≤ ∥Tυ (t)∥ · ∥ϕ∥Cτ + νN (υ, τ ) ∥Tυ (t − s)∥ · ∥us (ϕ)∥Cτ ds. 0
Thus, we can obtain the following local stability result of the trivial solution of (4). Theorem 3.1. If sup{Reµ : µ ∈ σ(Aυ )} = −α < 0, then the zero solution of (4) is locally asymptotically stable; If sup{Reµ : µ ∈ σ(Aυ )} = α > 0, then the zero solution of (4) is unstable. Proof . We start with the case where sup{Reµ : µ ∈ σ(Aυ )} = −α < 0. Then for each given ε ∈ (0, α], ) } {( there exists a constant M1 (ε) > 1 such that ∥Tυ (t)ϕ∥Cτ ≤ M1 (ε) exp 2ε − α t ∥ϕ∥Cτ for all t ≥ 0 and ϕ ∈ Cτ . It follows from (6) that ∥ut (ϕ)∥Cτ ≤M1 (ε) exp
∫ t ) } ) } {( ε − α t ∥ϕ∥Cτ + νN (υ, τ )M1 (ε) − α (t − s) ∥us (ϕ)∥Cτ ds. exp 2 2 0
{( ε
By applying the Gronwall’s inequality we have ∥ut (ϕ)∥Cτ ≤ M1 (ε) exp {(ε − α) t} ∥ϕ∥Cτ for all t ≥ τ . In what follows, we consider the case where sup{Reµ : µ ∈ σ(Aυ )} = α > 0. Using the same estimate as the above discussion and a similar argument to the usual functional differential equations (for instance, see Theorem 1.1, Chapter 10 in [13]), we are able to show that Eq. (4) has a nonzero solution u(x, t) which is defined for all t ≤ 0 and u(·, t) → 0 as t → −∞. Thus the zero solution is unstable. The proof is completed. □ Proof of Theorem 1.1. If sup{Reµ : µ ∈ σ(A)} = −α < 0, then there exists a sufficiently small υ > 0 such that sup{Reµ : µ ∈ σ(Aυ )} = − α2 < 0. This implies that for each small ε > 0, there exist a constant M (ε) and a neighborhood V (ε) ⊂ Cτ of the origin such that for ϕ ∈ V (ε), the solution v(t; ϕ, υ) of (4) is defined {( ) } for t ≥ 0 and ∥vt (ϕ, υ)∥Cτ ≤ M (ε) exp ε − α2 t ∥ϕ∥Cτ . Note that v(t; ϕ, υ) is continuous with respect to ) } {( υ, then we see that ∥vt (ϕ, 0)∥Cτ ≤ M (ε) exp ε − α2 t ∥ϕ∥Cτ , where v(t; ϕ, 0) is a solution of the following system with initial value ϕ: ⎧ ˙ =∆u(t) + f (x, u(t) + u∗ , u ˆt + u ˆ∗ ) − f (x, u∗ , u ˆ∗ ) , ⎨ u(t) ⎩
0=−
∂ u(t) + g (x, u ˆt + u∗ ) − g (x, u∗ ) . ∂n
which implies that the steady state solution u∗ of (1) is locally asymptotically stable. Similarly, we can show that the steady state solution u∗ of (1) is unstable if sup{Reµ : µ ∈ σ(A)} = α > 0. Thus, the proof of Theorem 1.1 is completed. □ CRediT authorship contribution statement Shangjiang Guo: Supervision, Conceptualization, Writing - review & editing, Funding acquisition. Shangzhi Li: Formal analysis, Writing - original draft.
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